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AF Geometry Unit 1
Unit Overview
Unit Title
Unit Designers
Coordinate geometry
Jennifer Tillotson, Sven A. Carlsson
Duration
IA Period
11 teaching days
1
Identify Desired Results
Standard
Previous Grade Level Standards / Previously Taught &
Related Standards
Use coordinates to prove simple geometric theorems algebraically.
G-GPE.4 Use coordinates to prove simple geometric theorems algebraically.
For example, prove that the triangle with vertices (1,5), (7,-1), and (-1,-3) is
isosceles.
G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and
use them to solve geometric problems (e.g., find the equation of a line
parallel or perpendicular to a given line that passes through a given point).
G-GPE.6 Find the point on a directed line segment between two given
points that partitions the segment in a given ratio.
G-GPE.7 Use coordinates to compute perimeters of polygons and areas of
triangles and rectangles, e.g., using the distance formula.
Relate work on parallel lines in G-GPE.5 to work on A-REI.5 in
Algebra I involving systems of equations having no solution or
infinitely many solutions.
G-GPE.7 provides practice with the distance formula and its
connection with the Pythagorean theorem. Building on their
exposure to the Pythagorean theorem in 8th grade as a way of
calculating distance, students use a rectangular coordinate system
to investigate and verify geometric relationships, primarily the
properties of special triangles and parallel/perpendicular lines.
G-CO.1 Know precise definitions of [angle, circle,] perpendicular line,
parallel line, and line segment, based on the undefined notions of point, line,
distance along a line, [and distance around a circular arc.]
In Units 7 and 8, students will return to the coordinate plane, using
the distance formula to examine the equation of a circle (U7) and
deepening their understanding of parabolas through exploring the
link between the equation of a parabola (focus/directrix) and its
graphical behavior (U8).
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AF Geometry Unit 1
Enduring Understandings and Big Ideas
Grade Level Enduring Understanding
Geometric and algebraic procedures are interconnected and build on one
another.
Two- and three-dimensional objects can be classified, described, and
analyzed by their geometric attributes using a variety of strategies, tools,
and technologies.
Congruent geometric figures can be mapped onto one another by one or
more rigid transformations (isometries). For similar figures, one of the
transformations need not be an isometry. These transformations can occur
on or off the coordinate plane.
Proof is an argument; it is a series of logically valid statements justified by
definitions, postulates, and theorems. Writing sound proofs develops
reasoning and justification skills.
Effective mathematical arguments involve both concise language and clear
reasoning, in the form of closely related steps justified by relevant
evidence.
All constructions are based on geometric properties of congruence.
Measurements (both direct and indirect) can be made to describe, compare,
and make sense of real-life objects. Geometric measurements can be
represented in algebraic expressions and equations.
What it looks like in this unit
Scholars will use their knowledge of the coordinate plane, slope (as
developed in Pre-Algebra and Algebra1), and algebraic methods to make
statements about geometric figures. Scholars will use algebraic
evidence to support arguments and classifications involving terminology
from Geometry.
Scholars will use coordinate geometry formulas (most notably the
distance and slope formulas) to classify triangles according to the
characteristics of their sides. Scholars will use rulers and protractors as
measuring tools, in the context of informally verifying side and angle
congruence.
The concept of isometry will be explored with isosceles triangles, as
congruent triangles will be provided on and off the coordinate plane,
with scholars commenting on the congruence of the triangles (using the
distance formula to evaluate the former and tools to investigate the
latter) – Isosceles, Congruent, Similar Worksheet.
Though formal proofs (2-column, paragraph, and flow) may be
introduced later in the course at teacher discretion, scholars will prove
statements about triangles in the Cartesian plane using slope and length
as evidence to both confirm and refute given statements about the
classification of the triangle.
Scholars will measure angles and side lengths using tools (ruler,
protractor) and algebraic formulas (distance), as well as using
measurements such as slope to draw conclusions about lines and sides.
Parallel Polygons Worksheet
Identify the Narrative
This unit reviews and extends scholars’ knowledge of linear equations, largely in the study of parallel and perpendicular lines on the coordinate plane.
A heavy emphasis is placed on point-slope form, as fluency with this form will be most useful during their Algebra 2 and Calculus classes.
Additionally, scholars will use other properties of coordinate geometry (the midpoint and distance formulas, equations of perpendicular bisectors, etc.)
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AF Geometry Unit 1
to solve problems. Scholars will be expected to determine which formulae are most appropriate in a given situation and synthesize their knowledge in
writing coordinate proofs. Writing proofs requires that scholars apply their knowledge of the properties and classification of shapes in a new way;
scholars will move from simply solving questions to articulating their arguments with precise terminology and explicitly stating their
reasoning/thought process. Coordinate proof thus deepens scholar knowledge (of both coordinate geometry and the properties of lines) while
strengthening their ability to articulate their ideas and support them with evidence.
Standards for Mathematical Practice
MP.1 Make sense of problems and persevere in Scholars will be required to analyze and decode problems on the coordinate plane in order to
solving them
understand what they are being asked to do.
Coordinate proofs require abstract reasoning skills (the ability to generalize or work with
generalizations) applied to coordinate geometry formulas (using quantitative evidence and
MP.2 Reason abstractly and quantitatively
computation to support their reasoning). Additionally, scholars will use their reasoning and
experience with the derivation of formulas to determine which formulas are to be used.
See: Midpoints, Perpendicular Bisectors, and Distance WS.
With the introduction of coordinate proof, scholars will routinely prove statements about geometric
figures and analyze the arguments put forth by others. They will be expected to use coordinate
MP.3 Construct viable arguments and critique
geometry formulas flexibly, as well as to work with multiple types of proof (two-column, paragraph,
the reasoning of others
flow, and coordinate).
See: Midpoints, Perpendicular Bisectors, and Distance WS.
MP.4 Model with mathematics
Scholars will use rulers and protractors to measure segments and angles respectively, in contexts
MP.5 Use appropriate tools strategically
where either (or neither) tool may be most appropriate. See: Parallelogram or not WS
Scholars will use precise geometric definitions and terminology when classifying triangles according
MP.6 Attend to precision
to their sides and when locating points of concurrency of triangles on the coordinate plane.
Scholars will articulate not only the meaning but the difference (or equivalence) of expressions such
MP.7 Look for and make use of structure
as 𝑥1 − 𝑥2 , 𝑥2 − 𝑥1 , or 𝑥1 + 𝑥2 in the distance, midpoint, and slope formulas.
MP.8 Look for and express regularity in
repeated reasoning
Major Misconceptions & Clarifications
Misconception
To show that a triangle is isosceles, slope can be
used.
Clarification
The terms isosceles, equilateral, and scalene all refer to the lengths of the sides of a triangle.
Slope can be used to determine whether sides are perpendicular or oblique; the slope of a
segment is unrelated to its length and thus does not need to be considered when classifying a
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AF Geometry Unit 1
The coordinates of the midpoint can be used as
(𝑥1 , 𝑦1 ) in the midpoint formula (when finding an
endpoint).
There is no fast way to check if a simplified
radical is correct.
triangle as isosceles. The importance of reading comprehension should be explicitly addressed –
have scholars regularly state in their own words what a question is asking and which ‘trigger
words’ in the question prompt tell them which formulas to use. Annotate the question!
The midpoint formula treats whatever points are inserted as endpoints and locates the midpoint
of the segment between them. The formula cannot directly output endpoints; the formula can
only be used indirectly to calculate an endpoint, and this will always require an equation being
set up. Scholars should red flag any question where they are given the midpoint, as its
coordinates must be handled differently from those of an endpoint. There are several possible
methods: scholars can label the midpoint (if given) as (𝑥𝑀 , 𝑦𝑀 ); alternatively, scholars can
sketch a rough diagram and label the known points with values/the missing point as (x, y). Build
a habit of checking answers for reasonableness.
To quickly check if a simplified radical is correct, find the product of the square of each of the
terms and compare to the original radicand. For example, to check that √50 = 5√2 has been
2
The perpendicular bisector of a segment passes
through one of the endpoints of the segment.
simplified correctly, simply multiply 52 by √2 , or 25 x 2. This equals 50, our original
radicand, so we have simplified correctly. Similarly, to check √48 = 4√3, we multiply 16 by 3
and arrive at 48, our original radicand.
Each word in the phrase “perpendicular bisector” has meaning. Perpendicular leads to a
consideration of slope, as perpendicular lines are lines with opposite reciprocal slopes. Bisector
leads to a calculation of the midpoint, as a bisector passes through the midpoint of the given
segment. A consistent theme of the course is this emphasis on vocabulary: what does each word
mean? How does that contribute to process? Make sure scholars are annotating the question and
using quick sketches to have a visual anchor alongside their work. As always, push scholars to
understand the process, rather than to see this as an algorithm to memorize.
Skills and Procedural Knowledge
Write the equation of a line given a slope and y-intercept, or given the slope and any point on the line, or given any two points on the line.
Identify line segments that are parallel or perpendicular given their slopes, points on the lines, or the equations of the lines.
Write the equation of a line parallel or perpendicular to a given line, passing through a given point.
Calculate the midpoint or length of a line segment, given its endpoints.
Use the Pythagorean Theorem to find missing sides of right triangles.
Describe relationships within triangles, and use these relationships to classify the shapes based on their characteristics.
Find the area and perimeter of triangles by applying formulas.
Find a point on a segment that partitions the segment into a given ratio.
Solve a system of linear equations.
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AF Geometry Unit 1
Write an equation of the perpendicular bisector of a segment, given its endpoints.
Use coordinate geometry formulas to prove statements about figures on the coordinate plane (for example, proving that a triangle is/isn’t a right
triangle, finding the perimeter of a triangle, etc.)
Unit Vocabulary
Slope: the slope m of a non-vertical line is the ratio of the vertical change (the rise) to the horizontal change (the run) between any two points on the
𝑦 −𝑦
line. The slope of a line can be calculated using the equation 𝑚 = 𝑥2 −𝑥1.
2
1
Y-intercept(s): the point(s) where a curve crosses the y-axis.
X-intercept(s): the point(s) where a curve crosses the x-axis.
Slope-intercept form: a form of a linear equation; 𝑦 = 𝑚𝑥 + 𝑏, where m is the slope of the line and b is the y-intercept of the line.
Standard form: a form of a linear equation, 𝐴𝑥 + 𝐵𝑦 = 𝐶, where A, B, and C are integers and A is positive.
Point-slope form: a form of a linear equation, 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ), where (𝑥1 , 𝑦1 ) is a point on the line and m is the slope of the line.
Length of a Segment: the distance between the endpoints of the segment
Parallel lines: two lines that do not intersect. Parallel lines have equal slopes, but different y-intercepts.
Perpendicular lines: two lines that intersect to form a right (90o) angle. Perpendicular lines have opposite reciprocal slopes.
Midpoint: a point that bisects a segment (divides it into two congruent segments). The midpoint of a line segment on the coordinate plane can be
𝑥 +𝑥 𝑦 +𝑦
calculated by 𝑀 = ( 1 2 2 , 1 2 2 ).
Perpendicular bisector: A segment or line that is perpendicular to a segment and bisects it.
Types of triangles – right (one right angle; two perpendicular sides), scalene (no congruent sides), isosceles (at least two congruent sides), equilateral
(all sides congruent).
Circumcenter: the point of intersection (concurrency) of the three perpendicular bisectors of the sides of any triangle. The circumcenter is the center
of a circle that is circumscribed about the triangle (passing through all three vertices), thus the circumcenter is equidistant from all three vertices. The
circumcenter of a triangle may lie inside the triangle (if the triangle is acute), on a side of the triangle (if the triangle is right), or outside the triangle (if
the triangle is obtuse).
Altitude: the perpendicular line segment from the vertex of a triangle to the opposite side. Scholars should not be exposed to altitudes until the Unit
Exam.
Incenter: the point of concurrency of the angle bisectors. The incenter is the center of a circle that is inscribed in the triangle (tangent to all three
sides). This will not be seen in any problems until Constructions in Unit 3.
Median: in a triangle, the median is the segment from the vertex to the midpoint of the opposite side.
Centroid: the point of intersection (concurrency) of the three medians of any triangle. The centroid always lies in the interior of a triangle. The
centroid is the center of mass of a triangle (the point at which a triangle can be balanced), and it is located 2/3 of the way from the vertex to the
midpoint.
Orthocenter: the point of intersection (concurrency) of the three altitudes of any triangle. What is interesting about the orthocenter is that if you
consider the orthocenter and the three vertices of the triangle (4 points total), any one of those four points is the orthocenter for the other three. The
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AF Geometry Unit 1
orthocenter of a triangle may lie inside the triangle (if the triangle is acute), on one side of the triangle (if the triangle is right), or outside the triangle
(if the triangle is obtuse).
Topics:
Topic 1 – Lines, slope and equation of
Topic 2 – Segments, length, midpoint, and perpendicular bisector of
Topic 3 – Triangles in the Cartesian plane
Aims Sequence
Content
Lesson #
Standards
1.1
G-GPE.5
T1
MPS
Aim/Exit Ticket
Key Points, Resources and Notes
MP.2
MP.7
SWBAT interpret slope to classify lines as parallel,
perpendicular, or neither;
SWBAT use the slope of a line to find coordinates of
points on the line.
1. A certain line passing through (0, -2) and (r, 4)
has a slope of 2. Find the value of r. Show all of
your work for full credit.
a) 1
b) 3
c) 5
d) 12
The slope of a line can be calculated by dividing the
𝛥𝑦
change in y by the change in x (𝑚 = 𝛥𝑥 ), or
2
1.2
T1
G-GPE.5
MP.7
2. A line with a slope of 5 passes through the
points (p, 3) and (10, p). Find the value of p.
3. The line containing the points (6, y) and (8, -2)
is perpendicular to the line containing the points
(-4, 1) and (3, -5). Find the value of y.
Given the equation of a line in any form, SWBAT find
the slope of a line parallel or perpendicular to the given
line;
Given the equations of two lines, SWBAT determine
whether the lines are parallel, perpendicular, or neither
by comparing their slopes.
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𝑦 −𝑦
equivalently by using the slope formula (𝑚 = 𝑥2 −𝑥1).
2
1
If the x- or y-coordinate of a point is unknown, it is
possible to use the slope formula to solve for this
missing value by substituting known quantities and
solving the resulting equation.
Note: this lesson can be shortened to allow for review
of linear equations, i.e. converting between forms or to
slope-intercept form. Teachers may also use this time to
review other selected concepts from Algebra – writing
linear equations given slope and a point, writing linear
equations from graphs, writing equations of vertical and
horizontal lines, etc.
To write the equation of a line, you need to know the
slope and a point on the line. If that point is not the yintercept, you can write the equation by substituting the
slope and point into point-slope form.
If you are given two points on a line, you must first find
the slope of the line between the two points using the
slope formula, and then you can use point-slope form to
AF Geometry Unit 1
1. Fill in each blank below.
A line parallel to y = ½x – 2 must have a slope of
____________.
A line perpendicular to y = 6x + 2 must have a slope of
____________.
A line parallel to 3x + 4y = 8 must have a slope of
____________.
2. Find the slope of a line perpendicular to the one
shown below:
–5
1
−
5
5
1
5
write the equation of the line. Either point can be used
as the point in point-slope form.
Parallel lines have the same slope; perpendicular lines
have negative reciprocal slopes. The product of a
number and its reciprocal is -1; the reciprocal of an
1
𝑏
𝑐
integer 𝑎 is 𝑎 and the reciprocal of 𝑐 is 𝑏.
Just because two lines intersect does not mean that they
are perpendicular.
3. The lines with equations y = 5x + 2 and y = –5x
– 3 are…
a) Parallel
b) Perpendicular
c) Segments
d) Intersecting, but not perpendicular
Justify your answer.
4. Describe the relationship between the set of
lines below as parallel, perpendicular, or
neither. Show your work or explain your
answer.
2x + 7y = 6 and 14x + 20 = 4y
1.3
G-GPE.5
Given the equation of a line and a point in the xy-plane,
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Same key points as above, with the following additions:
AF Geometry Unit 1
T1
SWBAT write the equation of a line parallel or
perpendicular to the given line passing through the
given point using point-slope form;
Write the equation of each line described below in (a)
slope-intercept form and (b) point-slope form.
1. Write the equation of a line passing through (2,
9) that is parallel to the line shown below.
The slope of a line is the coefficient of the x term when
the equation is in slope-intercept form. To calculate the
slope you need two points on the line.
Perpendicular lines have negative reciprocal slopes, not
reciprocal slopes.
To avoid arithmetic errors, ALWAYS write the formula
first, then substitute values
2. ⃡𝑌𝐴 passes through the points (6, -1) and (1, -2).
⃡𝑇𝑍 is perpendicular to ⃡𝑌𝐴, and passes through (5, -2). Write the equation of ⃡𝑇𝑍 in point-slope
form.
1.4
T1
Flex Day
1.5
T2
G-GPE.4

MP.1
MP.7
Questions involving variables/unknowns as coefficients
eg. Find the equation of the line through (4,-5) that is perpendicular to the line 𝑦 = 3𝑎𝑥 − 𝑏.
 Additional problems involving linear equations
e.g. Given the line 3𝑥 − 2𝑦 = 5, give the equation of a line perpendicular, parallel, oblique…
 Pre-teach/review of solving linear systems, in preparation for 1.9
Given two endpoints of a line segment, SWBAT
The x-coordinate of the midpoint of a line segment is
calculate the midpoint of the segment; given one
the average of the x coordinates of the endpoints; the
endpoint and the midpoint of the segment, SWBAT
same is true for the y-coordinate.
calculate the coordinates of the other endpoint.
The midpoint M of a segment with endpoints at (x1, y1)
𝑥 +𝑥 𝑦 +𝑦
Questions 1 and 2 refer to the triangle ABC pictured
and (x2, y2) is given by M = ( 1 2 2 , 1 2 2 ).
below.
To find a missing endpoint (if given the midpoint and
1. Use the diagram below to find point Q, the
other endpoint), substitute in known values, set up two
̅̅̅̅ . The segment ̅̅̅̅
midpoint of 𝐵𝐶
𝐴𝑄 is called a
equations (one for the x-coordinate, the other for the ymedian of triangle ABC.
coordinate), and solving each equation. Make sure to
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AF Geometry Unit 1
rewrite the final answer as an ordered pair.
When applying the slope, midpoint, or distance
formulas, it doesn’t matter which point is designated
(x1, y1) or (x2, y2). The reason for this is as follows:
In the slope formula, the magnitude of the difference
between the x- and y-coordinates will be the same –
only the sign will differ. However, because the sign will
change for both the numerator and denominator, the
end result will be the same.
In the midpoint formula, the answer remains the same
because addition is commutative.
In the distance formula, the result is the same because
(𝑥2 − 𝑥1 )2 = (𝑥1 − 𝑥2 )2 .
1.6
T2
G-GPE.4;
G-GPE.6
MP.1
̅̅̅̅ (not shown). What are
2. A is the midpoint of 𝐵𝑃
the coordinates of P? Is more than one answer
possible? Why or why not?
3. Segment ̅̅̅̅
𝐺𝑂 has an endpoint G with coordinates
̅̅̅̅, has coordinates
(2, -6). M, the midpoint of 𝐺𝑂
(-3, 1). What are the coordinates of the other
endpoint, O?
4. Segment ̅̅̅̅
𝑅𝑈 has endpoints R (-9, -1) and U (12, -5), and a midpoint M, (x, y). What is the
value of 2x – y?
Given two endpoints of a line segment, SWBAT
The distance formula is derived from the Pythagorean
calculate the length of the segment by using the
theorem.
distance formula;
When simplifying radicals by hand, show every step to
avoid errors, e.g. √12 = √4 ∙ 3 = 2√3. Also check
Given the endpoints of a line segment, SWBAT find the final answer.
coordinates of a point that partitions the segment in a
To partition a segment, find the change in the x- and ygiven ratio.
coordinates of the endpoints, set up a proportion with
Use the distance formula to answer each of the
the desired fraction, then apply it to each coordinate.
following questions. Simplify all radicals and also give Scholars are actually using similar triangles and the
answers as decimals rounded to the nearest tenth, if
ratio of similitude when applying this method.
necessary.
The distance formula can be used to verify that this
1. Explain why you can use the coordinates of
ratio is correct and the correct coordinate have been
either point as (x2, y2) when finding the distance determined.
between two points.
2. Consider the points F (-4, 3) and G (2, 5). Find
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AF Geometry Unit 1
the length of ̅̅̅̅
𝐹𝐺 in simplest radical form. How
long is a segment twice as long as ̅̅̅̅
𝐹𝐺 ?
3. Given the points A (-4, 2) and B (2, 8), find the
coordinates of point P on directed line segment
3
̅̅̅̅
𝐴𝐵 that is 4 of the way from A to B. [The use of
the grid below is optional.]
4. Find the coordinates of the point 1/3rd of the
way from B to A.
1.7
T2
1.8
T3
Flex Day
G-GPE.4;
G-GPE.7
MP.1
MP.2
MP.3
Further work with partitioning a segment into 1/3, 2/3, 4/5, etc.
Informally confirming partitions by using a ruler and setting up proportions
Given the coordinates of a triangle on the coordinate
Same key points as above, with the addition of:
plane, SWBAT name the polygon and find the
A scalene triangle has no congruent sides, an isosceles
perimeter/area of the figure.
triangle has (at least) two sides congruent, and an
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AF Geometry Unit 1
MP.6
1.9
T2
G-GPE.5
Use ΔABC with coordinates A(1, 6), B(2, 9), and C(7,
equilateral triangle has all sides congruent.
10) to answer questions 1 and 2.
1. Find the exact perimeter of ΔABC.
2. Which of the following is true?
i.
ΔABC is scalene
ii.
ΔABC is isosceles
iii.
ΔABC is equilateral
iv.
Not enough information is given to answer
the question
3. Explain how you found your answer to Q2
above.
4. Your friend Barack tells you he’s going to prove
that the triangle OBA below is not equilateral,
̅̅̅̅,
and he starts by finding the slopes of sides 𝑂𝐵
̅̅̅̅ , and 𝑂𝐴
̅̅̅̅. Explain to him why he is doing
𝐴𝐵
extraneous (unnecessary) work/Explain to him
why his approach will not work in his proof.
Given two endpoints of a line segment, SWBAT write
the equation of the perpendicular bisector of the
segment.
1. Which of the following must be calculated in
order to write an equation of the perpendicular
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Annotate the question! The words “perpendicular
bisector” tell you what to do:
Perpendicular implies slope; specifically, find the
negative reciprocal slope.
Bisector implies splitting into two equal pieces;
AF Geometry Unit 1
i.
ii.
iii.
bisector of a segment?
The length of the line segment
The midpoint of the line segment
The slope of the line segment
A.
B.
C.
D.
E.
i, ii, and iii
i and ii only
ii and iii only
i and iii only
None of the above
2. Write the equation of the perpendicular bisector
̅̅̅̅, whose endpoints are L(-4,3)
of segment 𝐿𝑀
and M(2, 5).
a. 𝑦 = −3𝑥 + 1
1
13
b. 𝑦 = 3 𝑥 + 3
c. 𝑦 = −3𝑥 + 11
d. 𝑦 = 𝑥 + 5
e. None of the above
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specifically, find the midpoint).
Emphasize to scholars that it is important to understand
the process and not simply try to memorize an
algorithm. Scholars should sketch a quick visual anchor
to check the reasonableness of their final answer and to
guard against errors. Clearly label all steps with headers
so that you know what you are calculating
Note: make sure to teach perpendicular bisectors both
graphically and algebraically – for example, by giving
scholars the graph of a segment and asking them to
write the equation of the perpendicular bisector of the
segment. This can also be useful in having them
visualize their errors, e.g. using an endpoint of the
segment instead of the midpoint to write the equation.
AF Geometry Unit 1
1.10
T3
G-GPE.4
MP.1
Given the coordinates of the vertices of a triangle,
SWBAT write the equations of the medians and
perpendicular bisectors;
Given the coordinates of the vertices of a triangle,
SWBAT find the coordinates of the circumcenter and
centroid of the triangle.
-----------------------------------------------------------------Triangle ABC has vertices A (3, 3), B (7, 9), and
C (11, 3). Determine the point of intersection of the
medians, and state its coordinates. [The use of the set of
axes below is optional.]
The vertices of ΔJKL are J (-2, 0), K (2, 8),and L (7,
3). Find the coordinates of the point that is equidistant
from J, K, and L. [The use of the set of axes below is
optional.]
1.11
Flex Day
This is an exciting lesson as it combines ideas from 1.4
(midpoints), 1.5 (ratios), and 1.8 (perpendicular
bisectors). The points of concurrency in a triangle are of
interest to us because of their unique properties.
The circumcenter of a triangle can be found by finding
the intersection point of any two perpendicular
bisectors of the sides of the triangle. This involves
writing the equations of both perpendicular bisectors
and solving a linear system.
Similarly, the centroid of a triangle can be found by
calculating the intersection point of any two of the
medians. However, the centroid can also be found by
simply finding the endpoints of one of the medians,
then partitioning that segment into the ratio 2:1 using
the slope partition method of 1.5.
As always, emphasis should be placed on understanding
how the process for (finding the circumcenter) flows
naturally from the definition, rather than presenting this
aim as a process to be memorized. While scholars are
memorizing that the centroid is located 2/3rd of the way
from the vertex to the midpoint, they are not
memorizing an algorithm that spits out the centroid –
they are applying a method learned in 1.5 to a definition
(median) learned today.
NOTE: Do not mention altitudes and orthocenters, as
these terms should be first introduced to students on the
Unit Exam.
Areas of triangles in the coordinate plane
For added difficulty, consider making no side of the triangle horiztonal/vertical. This requires scholars to find the
equations of the lines serving as the height and base, to solve a system to find their point of intersection, and lastly
Page 14 of 27
AF Geometry Unit 1
to apply the distance formula to determine the height of the triangle.
Unit Exam
1.
Find the y-intercept of the line passing through (-4, 2) and (6, 6).
(A)
18
5
(B) −
12
5
(C) −6
(D) −
18
5
(E) 18
2.
Which of the following lines is perpendicular to the line 𝑥 + 4𝑦 = −24 and passes through the
point (5, -2)?
(A) 𝑦 = 4𝑥 + 22
1
(B) 𝑦 = 4 𝑥 − 22
1
(C) 𝑦 = − 4 𝑥 − 22
(D) 𝑦 = 4𝑥 − 22
(E) 𝑦 = −4𝑥 + 18
3.
Line 𝑙 has a positive slope and passes through the origin. If line 𝑘 is perpendicular to line 𝑙, which
of the following must be true?
(A) Line k passes through the origin.
(B) Line k has a positive slope.
(C) Line k has a negative slope.
(D) Line k has a positive x-intercept.
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AF Geometry Unit 1
(E) Line k has a negative y-intercept.
4.
A triangle has vertices at A (-4, 1), B (-8, -3), and C (-6, -5).
Which of the following is true about triangle ABC?
̅̅̅̅ is perpendicular to 𝐴𝐶
̅̅̅̅ .
(A) ∆𝐴𝐵𝐶 is a right triangle because 𝐴𝐵
(B) ∆𝐴𝐵𝐶 is isosceles triangle because ̅̅̅̅
𝐴𝐵 is perpendicular to ̅̅̅̅
𝐴𝐶 .
̅̅̅̅ is perpendicular to 𝐵𝐶
̅̅̅̅ .
(C) ∆𝐴𝐵𝐶 is a right triangle because 𝐴𝐵
̅̅̅̅ is perpendicular to 𝐵𝐶
̅̅̅̅ .
(D) ∆𝐴𝐵𝐶 is isosceles because 𝐴𝐵
(E) ∆𝐴𝐵𝐶 is not a right triangle.
5.
Which equation below represents the perpendicular bisector of the line segment connecting the
points (3, -1) and (-9, 5)?
(A) 𝑦 = −2𝑥 + 8
(B) 𝑦 = 2𝑥 − 8
(C) 𝑦 = −2𝑥 − 4
(D) 𝑦 = 2𝑥 + 8
(E) 𝑦 = 2𝑥 − 4
6.
A line passing through the points (a, -5) and (2, -6) is parallel to the line represented by the
3
equation 𝑦 = (𝑎) 𝑥 + 2. Find the value of a. Show all of your work for full credit.
(A) −1
(B)
(C)
3
4
3
2
(D) 3
Page 16 of 27
AF Geometry Unit 1
7.
ΔACD has vertices at A (-2, 4), C (6, 7), and D (-7, -5). Which of the following formulas would be
needed to classify the triangle as scalene, isosceles, or equilateral?
(A) The midpoint formula
(B) The distance formula
(C) The slope formula
(D) Both (B) and (C)
(E) (A), (B), and (C)
8.
̅̅̅̅ is M (-10, -16). If one endpoint is B (-1, 8), what are the coordinates of the
The midpoint of 𝐵𝐶
other endpoint, C?
(A) (−21, −40)
(B) (−20, −40)
(C) (−19, −40)
(D) (−21, −24)
(E) (8, 32)
9.
A quadrilateral is a shape with four sides. In the quadrilateral GRIT below, two triangles are
̅̅̅ .
formed by constructing 𝐺𝐼
a) Prove that both triangles are right triangles.
Page 17 of 27
AF Geometry Unit 1
10.
(b) Prove that ∆𝐺𝑇𝐼 is not equilateral.
Write the equation of the line that passes through the point P (1, -2) and is parallel to
the line shown in the diagram below.
Page 18 of 27
AF Geometry Unit 1
11.
The map below shows a straight local highway that runs between two towns in Massachusetts.
Highway planners want to build a rest stop somewhere between the two towns.
If the lead planner wants the rest stop to
be 1/3 of the way from Ashton to Bedford,
at what coordinates will the town be
located?
Page 19 of 27
AF Geometry Unit 1
12.
Consider ∆𝐴𝐵𝑂 below.
(a) Find the perimeter of the
triangle.
(b) Find the coordinates of the centroid.
13.
(c) Your friend Barack wants to find the area of the triangle. He says to you, “Let’s say ̅̅̅̅
𝐴𝐵 is the
base. AB is 7 and from B to O is 5 units up, which is the height, so the area should be about
½ x 7 x 5 or 17.5 units2.” Do you agree or disagree? Explain your reasoning.
Consider the triangle ABC with vertices A(1,0), B(5,8), and C(10,3).
An altitude of a triangle is a segment drawn from a vertex to the opposite side that is also
perpendicular to the side.
̅̅̅̅ .
a. Write the equation of the altitude to side 𝐴𝐶
̅̅̅̅ .
b. Write the equation of the altitude to side 𝐵𝐶
Page 20 of 27
AF Geometry Unit 1
c. The orthocenter is the point of concurrency of the altitudes. In other words, the orthocenter is
where the altitudes of a triangle intersect. Find the coordinates of the orthocenter of ∆𝐴𝐵𝐶.
̅̅̅̅ is parallel to the altitude to side
d. Your friend Barack says to you, “The perpendicular bisector of 𝐴𝐶
̅̅̅̅
𝐴𝐶 .” Decide whether he is correct or incorrect and then prove your answer.
Unit Exam Scoring Guide
Question Correct Answer
1
A
Points
2
2
D
2
3
4
5
C
C
D
2
2
3
6
D
3
7
8
C
C
2
2
9a
∆𝐺𝑇𝐼 is a right
triangle because
segments GT and TI
are perpendicular.
We know the
segments are
3
Point Breakdown
1 – Linear equation with slope of 2/5
1 – Answer
1 – Indicate desired slope is 4 or 4⁄1
1 – Answer
2 – Answer
2 – Answer
1 – Correct midpoint (-3,2)
(lose ½ point for each incorrect coordinate)
1 –Indicate the desired slope is 2
1 – Answer
−5−(−6)
1
1 – Use given points to calculate slope 𝑎−2 = 𝑎−2; simplification not necessary
1 – Equate the slopes of both lines
1 – Answer
2 – Answer
1 – Clear indication of method used, be it graphical or algebraic
1 – Answer
1 – Calculate slopes as -2 and ½ (do not need to simplify).
1 – Conclude perpendicularity by comparing slopes of adjacent sides
1 – Answer
Note: if scholar calculates area of a larger rectangle then subtracts area of triangles,
score as follows:
Page 21 of 27
AF Geometry Unit 1
9b
10
11
perpendicular
because their slopes
are opposite
reciprocals; 𝑚̅̅̅̅
𝐺𝑇 =
−2 and 𝑚 𝑇𝐼
̅̅̅ = 1/2.
Similarly, since
𝑚𝐺𝑇
̅̅̅̅ = 𝑚𝑅𝐼
̅̅̅ and
𝑚 𝑇𝐼
=
𝑚
̅̅̅̅ , we
̅̅̅
𝐺𝑅
know that ̅̅̅
𝑅𝐼 ⊥ ̅̅̅̅
𝐺𝑅,
so ∆𝐺𝑅𝐼 is right.
GT = 2√5 and TI
=3√5, so 𝐺𝑇 ≠ 𝑇𝐼.
In an equilateral
triangle, all sides are
congruent, so ∆𝐺𝑇𝐼
is not an equilateral
triangle because its
two sides GT and TI
are not congruent.
1
𝑦 + 2 = − (𝑥 − 1)
2
1
(−1, − )
3
1 – Correct area for larger rectangle
1 – Correct area for at least 3 of 4 triangles
1 - Answer
2
1 – Argument: if all sides are not congruent, then the triangle is not equilateral.
1 – Evidence that two sides are not congruent through a distance computation (correct
or incorrect).
3
1 – Indicate the slope of the given line is − 2
1 – Scholar uses parallel lines have equal slopes
1 – Consistent Answer (for eligible scholars)
Eligibility: scholar indicates negative slope for the graphed line
5
1
Note: For example, a scholar who incorrectly identifies the slope of the graphed line
2
as −2⁄5, then goes on to give the consistent answer 𝑦 + 2 = − 5 (𝑥 − 1) or correctly
simplifies to 𝑦 = −25𝑥 − 85, will earn a 0-1-1, or 2 points total.
1 – Calculate the slope between Ashton and Bedford
1 – Find correct slope of 5⁄6
1 – Multiply ∆𝑥 and ∆𝑦 by 1⁄3
1 – Add ∆𝑥/3 and ∆𝑦/3 to coordinates of Ashton
1 – Answer
Note: If scholars using proportions, can earn the first four points through correctly setPage 22 of 27
AF Geometry Unit 1
up proportions
𝑥−(−3)
1
= 3 and
6
𝑦−(−2)
5
1
= 3.
1
Note: If scholar gives an answer of (1, 3), the point 1/3rd of the way from Bedford to
Ashton, earns a maximum of 1-1-1-0-0, or 3 points total.
7 + √85 +
√106 units
(5,2)
2
1 – Correct value for either AO or BO
1 – Answer
3
12c
Barack is incorrect.
When calculating
the area, the base
and height must be
perpendicular. If
̅̅̅̅
𝐴𝐵 is used as the
height, then Barack
must use a
horizontal distance
as the height, not a
vertical distance.
2
1 – calculate two correct median equations (y=1.33x-4.67, y=0.17x+1.17, y=-x+7)
1 – Solve linear system formed by scholar’s median equations
1 – Answer
1 – Answer (Barack is incorrect)
1 – In the area formula, the base and height must be perpendicular.
13a
𝑦 = −3𝑥 + 23
3
12a
12b
13b
13c
𝑦 =𝑥−1
(6,5)
3
3
1 – Find the slope of ̅̅̅̅
𝐴𝐶 as 1/3
1 – Find the opposite reciprocal of their 𝑚̅̅̅̅
𝐴𝐶
1 – Answer
Note: incorrect final answer earns a maximum of 0-1-0, or 1 point total
1 – Find the slope of ̅̅̅̅
𝐵𝐶 as −1
1 – Find the opposite reciprocal of their 𝑚𝐵𝐶
̅̅̅̅
1 – Answer
Note: incorrect final answer earns a maximum of 0-1-0, or 1 point total
2 – attempt to solve a linear system composed of linear equations from (a) and (b)
using either Substitution or Linear Combination
Page 23 of 27
AF Geometry Unit 1
1 – Answer
Note: 1/3 if correct point of intersection is found graphically with no evidence that
scholar checked the point of intersection in both equations.
13d
Barack is correct.
The slopes of
parallel lines are
equal. The slope of
the altitude is −3,
and the slope of the
perpendicular
bisector is the
opposite reciprocal
of 1/3, which is -3 as
well.
2
Finding the correct point of intersection from a labeled graph with evidence of scholar
checking the solution in both equations earns 3/3.
1 – Answer
1 – Evidence
Appendix A: Tasks
Performance Assessments
Page 24 of 27
AF Geometry Unit 1
Triangles inscribed in a circle (link:
Illustrative Mathematics)
Suppose A = (−1, 0) and B = (1, 0) are points in the
coordinate plane as pictured at right.
Suppose C = (x, y) is a third point in the coordinate
plane, and that C does not lie on the x-axis.
𝑦
1−𝑥
Show that if ∠𝐴𝐶𝐵 is a right angle, then 𝑥+1 = 𝑦 .
Solve the above equation to show that if ∠𝐴𝐶𝐵 is a
right angle, then C (x, y) must lie on 𝑥 2 + 𝑦 2 = 1.
Getting to know Perpendicular Bisectors
Consider the line segment with endpoints A (-1, 2) and B (3, -4).
̅̅̅̅ . Call this line 𝒍.
a. Write the equation of the perpendicular bisector of 𝐴𝐵
b. Prove that the point (0, 5) is not on the line 𝒍.
c. List one point on the line l that is in the first quadrant and one point on l that is in the third quadrant.
d. Find a new point that is on the line l (it may lie in any quadrant). Name the point P. Prove that P is
equidistant from A and B.
e. Is the point F (2017, 1342) equidistant from points A and B? What does this tell you about the point F
with respect to the line l ?
Applications of the Circumcenter
Your family is considering moving to a new home. The diagram shows the locations of the office and
factory where your parents work, and the location of your school. The three locations form a triangle.
Your family wants to live at a point that is equidistant from each location. Describe how you could use
the diagram to find a point that fits this criteria, citing any formulas, postulates, or theorems you use.
Page 25 of 27
AF Geometry Unit 1
Critiquing vague language
Taken from Exeter Math 1, page 47
Working with Standard Form
Taken from Exeter Math 1, Page 50
Appendix B: Teacher Background Knowledge
Important formulas, postulates, and theorems
𝑦 −𝑦
Slope formula: 𝑚 = 𝑥2 −𝑥1
2
1
Forms of linear
Slope-intercept form: 𝑦 = 𝑚𝑥 + 𝑏
equations
Standard form: 𝐴𝑥 + 𝐵𝑦 = 𝐶
Point-slope form: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Page 26 of 27
AF Geometry Unit 1
Slopes of lines on the
coordinate plane
Parallel lines have the same slope. Perpendicular lines have opposite (negative) reciprocal slopes.
Any line parallel to the x-axis has a slope of zero.
Any line parallel to the y-axis has an undefined slope.
𝑥1 +𝑥2 𝑦1 +𝑦2
Midpoint formula: 𝑀 = (
Coordinate geometry
formulas
Distance formula: 𝑑 =
,
)
2
√(𝑥2 − 𝑥1 )2 +(𝑦2 −
𝑥1 +𝑥2 +𝑥3 𝑦1 +𝑦2 +𝑦3
Centroid formula: 𝑀 = (
2
3
,
3
𝑦1 )2
) If teaching to scholars, relate this formula to the centroid as center of mass.
Questions The Teacher Should Be Able to Answer
1. Why are the slopes of perpendicular lines opposite reciprocals? or Why is the product of the slopes of perpendicular lines equal to -1?
While scholars have already learned that the product of the slopes of perpendicular lines is negative one, they may not be clear on why this is so. It is
quick to explain and stems from taking a segment with a given slope, say 𝑎/𝑏, rotating it 900 about an endpoint, showing that the horizontal and
vertical components have thus switched, and concluding that the new slope is −𝑏/𝑎.
See the following explanations for a quick refresher:
Central Oregon Community College
Drexel’s Ask Dr. Math
Page 27 of 27