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GEOMETRY
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
Lesson 13
M4
Period: _____Date______________
Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means
Learning Targets


Using coordinates, I can find the intersection of the medians of a triangle that meet at a point that is two-thirds
of the way along each median from the intersected vertex.
Using coordinates, I can prove a quadrilateral to be a parallelogram showing that diagonals of a parallelogram
bisect one another
̅̅̅̅ with 𝐵(−2, −1) and 𝐶(4, 1).
Opening Activity Line 𝐴 is the perpendicular bisector of segment 𝐵𝐶
What is the midpoint of ̅̅̅̅
𝐵𝐶 ?
̅̅̅̅ ?
What is the slope of 𝐵𝐶
̅̅̅̅ .)
What is the slope of line 𝐴? (Remember, it is perpendicular to 𝐵𝐶
̅̅̅̅ .
Write the equation of line 𝐴, the perpendicular bisector of 𝐵𝐶
Remember from Module 1……
All triangles have three medians will intersect at one point in the interior of the triangle called
the centroid or the point of concurrency.
The centroid of a triangle divides the medians into a 2:1 ratio
Example 1
Given triangle 𝑨𝑩𝑪 with vertices𝑨(𝟏, 𝟏),𝑩(𝟓, 𝟐), and 𝑪(𝟑, 𝟓), find the coordinates of the point of concurrency.
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Find point M midpoint of segment AB
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Find point N midpoint of segment CB
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Find point R midpoint of segment AC
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Draw all three medians and label the point concurrency D
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Use the ratio to find the coordinates of the point of concurrency
GEOMETRY
Lesson 13
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
M4
Period: _____Date______________
Quadrilaterals Properties
(Polygons having four sides)
Parallelograms
Trapezoids
2 sets of parallel sides
2 sets of congruent sides
Opposite angles congruent
Diagonals bisect each other
Kites
Have exactly one pair of opposite
sides that are parallel
2 pairs adjacent congruent sides
Opp. sides not congruent or parallel
2 pairs of opposite sides parallel
Opposite sides and opposite angles
are congruent
Rectangles
Rhombuses
Diagonals bisect each other
All of the properties of the
parallelogram PLUS
4 right angles
diagonals are congruent
All of the properties of the
parallelogram PLUS
4 congruent sides
diagonals bisect angles
diagonals are perpendicular
Isosceles Trapezoid
One set of parallel sides
Base angles and legs congruent
Diagonals are congruent
Opposite angles are supplementary
Squares
I have it all!
Coordinate Proofs
On a parallelogram (including rectangles, squares and rhombuses) diagonals that bisect each other.
On the coordinate plane, diagonals have the same midpoints, therefore they intercept each other.
A parallelogram is always a trapezoid, with two sets of opposite sides parallel.
On the coordinate plane we prove to sides to be parallel if they have the same slope
A rectangle is a parallelogram with all four right angles.
On the coordinate plane we prove that by showing consecutive sides have slopes that are opposite reciprocals
A rhombus is a parallelogram with all four sides equal. On the coordinate plane we prove that by using the distance
formula to find the length of all sides
GEOMETRY
Lesson 13
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
Name________________________
Period: _____Date______________
Example 2. Quadrilateral MATH with vertices M(3,-4) , A(0,2) , T(6,2) and H(9,-4)
A. Prove that Quadrilateral MATH is a parallelogram
Explanation: Quadrilateral MATH is a parallelogram
because two sets of opposite sides have the same
slope, meaning that two sets of opposite sides are
_________________.
B. Prove that Quadrilateral Math is a parallelogram using
diagonals
i.
What is the midpoint of segment MT
ii.
What is the midpoint of segment AH
C. Is quadrilateral MATH a trapezoid? Explain your
answer
Explanation: Quadrilateral MATH is a parallelogram
because the diagonals have the same midpoint,
therefore the diagonals of a parallelogram bisect one
another
D. Is quadrilateral MATH a rhombus?
E. Is quadrilateral MATH a rectangle? Explain your answer
GEOMETRY
Lesson 13
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
M4
Period: _____Date______________
Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means
Classwork
1. Let 𝐴(−23, 12), 𝐵(13, 36), and 𝐶(23, −1) be vertices of a triangle.
̅̅̅̅ and label M
a) Find the midpoint for segment 𝑨𝑩
b) Find the midpoint for segment ̅̅̅̅
𝐵𝐶 and label P
̅̅̅̅ and label N
c) Find the midpoint for segment 𝐴𝐶
d) Where will the medians of this triangle intersect? (point of concurrency or centroid)
2. Given quadrilateral 𝑱𝑲𝑳𝑴 with vertices 𝑱(−𝟒, 𝟐), 𝑲(𝟏, 𝟓), 𝑳(𝟒, 𝟎), and 𝑴(−𝟏, −𝟑):
a. Is it a trapezoid? Explain
b. Is it a parallelogram? Explain
c. Is it a rectangle? Explain
d. Is it a rhombus? Explain
e. Is it a square? Explain
f.
Find the point of interception of diagonals of
𝑱𝑲𝑳𝑴
GEOMETRY
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 13
M4
Name________________________
Period: _____Date______________
3. Given a quadrilateral with vertices 𝑬(𝟎, 𝟓), 𝑭(𝟔, 𝟓),𝑮(𝟒, 𝟎), and 𝑯(−𝟐, 𝟎):
a. Prove quadrilateral 𝑬𝑭𝑮𝑯 is a parallelogram.
b. Prove (𝟐, 𝟐. 𝟓) is a point on both diagonals of the quadrilateral.
4. Prove quadrilateral 𝑾𝑿𝒀𝒁 with vertices 𝑾(𝟏, 𝟑), 𝑿(𝟒, 𝟖), 𝒀(𝟏𝟎, 𝟏𝟏), and 𝒁(𝟒, 𝟏) is a trapezoid. Explain your
answer.
GEOMETRY
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
Lesson 13
M4
Period: _____Date______________
Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means
Homework
1) Prove that the quadrilateral whose vertices are the points A(-1,1), B(-3,4), C(1,5) and D(3,2) is a parallelogram.
2) Quadrilateral DEFG has vertices at D(3,4), E(8,6), F(9,9) and G(4,7). Prove that DEFG is a parallelogram using diagonals
GEOMETRY
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 13
M4
Name________________________
Period: _____Date______________
3) Quadrilateral ABCD has vertices 𝐴(2, 3), 𝐵(10, 3), 𝐶(10, −1), 𝑎𝑛𝑑 𝐷(2, −1).
Prove quadrilateral ABCD is a rectangle
4) Quadrilateral QRST has vertices 𝑄(6, 7), 𝑅(11, 7), 𝑆(8, 3), 𝑇(3, 3). Prove
quadrilateral QRST is a rhombus
5) The coordinates of the vertices of quadrilateral ABCD are
𝐴(4,1), 𝐵(1,5), 𝐶(−3,2) 𝑎𝑛𝑑 𝐷(0, −2). Prove the quadrilateral is a square.