Download 2012 geometry study guide

End of Course Geometry Study Guide
End-of-Course Exam Items
Logical Arguments & Proofs
Two & Three Dimensional Figures
Coordinate Geometry & Measurement
November, 2011
Estimated Number of Questions (37 questions)
6-8
24-26
7-9
Format of exam questions: 29 multiple choice, 5 completion items and 3 short answer questions. A
graphing calculator can be used for all questions and will be cleared before the exam. A formula sheet
and graph paper will be provided in the test booklet.
Study Guide Organization:
On the next page is the Review Topic Index. These items are key “green” standards that will be assessed
for graduation. The number of test questions for each topic heading is shown to help guide your study
priorities. Following the Index are Sample Questions for each topic. After each set of questions, there
are STUDY NOTES that explain the answers.
1
Review Topic Index:
(G1) Logical Arguments and Proofs
(6-8 points)
G1C Use deductive reasoning to prove that a valid geometric statement is true.
G1D Write the converse, inverse, and contrapositive of a valid proposition and determine their validity.
G1E Identify errors or gaps in a mathematical argument and develop counterexamples to refute invalid
statements about geometric relationships.
G1F Distinguish between definitions and undefined geometric terms and explain the role of definitions,
undefined terms, postulates (axioms), and theorems.
(G3) Two & three dimensional figures
(24-26 points)
G3A Know and apply basic postulates and theorems about triangles and the special lines, line segments,
and rays associated with a triangle.
G3B Determine and prove triangle congruence and other properties of triangles.
G3C Use the properties of special right triangles (30°–60°–90° and 45°–45°–90°) to solve problems.
G3D Know, prove, and apply the Pythagorean Theorem and its converse.
G3E Solve problems involving the basic trigonometric ratios of sine, cosine, and tangent.
G3G Know, prove, and apply theorems about properties of quadrilaterals and other polygons.
(G4)(G6) Coordinate Geometry & Measurement
(7-9 points)
G4B Determine the coordinates of a point that is described geometrically. Example: Determine the
coordinates for the midpoint of a given line segment.
G4C Verify and apply properties of triangles and quadrilaterals in the coordinate plane.
G6E Use different degrees of precision in measurement, explain the reason for using a certain degree of
precision, apply estimation strategies to obtain reasonable measurements for a given purpose.
G6F Solve problems involving measurement conversions within and between systems, including those
involving derived units, analyze solutions for reasonableness and appropriate units.
2
(G1) Logical Arguments and Proofs
G1C Use deductive reasoning to prove that a valid geometric statement is true
Sample item for Performance Expectation G.1.A/ M1.4.A
Which statement illustrated inductive reasoning?
O A.
John read “
” and then concluded that
O B.
John read “
O C.
John read the sequence “2, 5, 8, 11…” and concluded that the next term in the
” and then concluded that
.
.
sequence must be 14.
O D.
John knew that any number greater than
a true statement and concluded that
would make the inequality
would make the inequality
a true statement.
Answer: C
Sample item for Performance Expectation G.1.D/M2.3.C
Determine the contrapositive of the given statement.
If two angles are right angles, then they are congruent.
Which is the contrapositive of this statement?
O A.
If two angles are congruent, then they are right angles.
O B.
If two angles are not congruent, then they are not right angles.
O C.
If two angles are not right angles, then they are not congruent.
O D.
If two angles are not right angles, then they could be congruent.
Answer: B
3
Sample Question G1D1:
Sample item for Performance Expectation G.1.D/M2.3.C
The given statement is a valid geometric proposition.
If two angles are right angles, then they are congruent.
Write the converse of the given statement.
Determine whether the converse is valid or invalid and provide evidence (example or
counterexample) for your answer.
Is the converse of the statement valid or invalid? _______________________________________
4
2-point response: The student shows understanding of writing the converse of a valid proposition and
determining its validity by doing the following:
 Writes If two angles are congruent, then they are right angles, or equivalent, for the converse.
 Writes invalid with a counterexample.
1-point response: The student does one of the following:
 Writes If two angles are congruent, then they are right angles, or equivalent, for the converse.
 Writes If two angles are not right angles, then they are not congruent (the inverse) AND writes
invalid with counterexample.
 Writes If two angles are not congruent, then they are not right angles (the contrapositive) AND
writes valid with explanation.
5
Sample Question G1D3:
Answer:
Inverse: If the table top is not rectangular, then the diagonals are not congruent.
Contrapositive: If the diagonals are not congruent, then the table top is not rectangular.
6
Sample item for Performance Expectation G.1.E/ M2.3.B
Christine knows that if two polygons are congruent, then they must have the same
perimeter and area. She concludes that it is also true that if two polygons have the same
perimeter and area, then they are congruent.
Which pair of polygons can be used as a counterexample to Christine’s conclusion?
O A.
O B.
O C.
O D.
.
Answer: D
7
Sample item for Performance Expectation G.2.A/ M1.4.E
Line MN is the perpendicular bisector of line segment PQ at point R.
Which statement must be true?
O A.
O B.
O C.
O D.
Answer: D
Sample item for Performance Expectation G.2.B/M1.4.F
In the figure shown, lines a and b are parallel and line c is a transversal.
117.3o
a
xo
b
c
Determine the value of x.
Write your answer on the line.
What is the value of x? _________________
Answer: 62.7
8
Sample item for Performance Expectation G.7.B (G.3.A)/M2.6.B (M2.3.E)
The smallest angle of a triangle is three-fourths the measure of the middle angle, and the
middle angle is four-fifths the measure of the largest angle.
Determine the measure of the largest angle of the triangle.
Show your work using words and/or numbers.
What is the measure of the largest angle of the triangle? _______________ degrees
2-point response: The student shows understanding of solving a problem by doing the following:
Understanding:
3
4
 Uses
and
4
5
Strategy:
 Shows a strategy to determine the measure of the largest angle
Answer:
 Writes 75.
1-point response: The student does one of the following:
3
4
 Uses
and
and shows a strategy that could lead to the measure of the largest angle.
4
5
 Writes 75.
9
Sample item for Performance Expectation G.3.B/M2.3.F
A proof is shown.
Fill in the blanks for steps 4 and 5 to complete the proof.
Given: WY is the perpendicular bisector of XZ
Prove: WXY  WZY
Statements
Reasons
1. WY is the perpendicular bisector of XZ .
1. Given
2. Perpendicular lines form 90 degree
2. WYX  WYZ
angles
3. Reflexive property of congruence
3. WY  WY
4. A bisector divides a segment into two
4.
equal halves
5. WXY  WZY
5.
2-point response: The student shows understanding of proving triangle congruence by doing the
following:
 Writes XY  YZ , or equivalent, for statement 4
 Writes Side-Angle-Side, or equivalent, for reason 5
Students can earn 1 point by doing one of the above.
10
Sample Question G3A1:
Answer: B
Sample item for Performance Expectation G.3.C,D/M2.3.G, I
In rectangle ABCD, the length of AC is 4 and
.
A
B
D
C
Determine the exact length of side CD.
Write your answer on the line.
What is the exact length of side CD? _________________
Answer:
11
Sample item for Performance Expectation G.3.C,D/M2.3.G, I
Square ABCD is shown. Diagonal BD has a length of 10 inches.
A
B
D
C
Determine the exact length of a side of the square.
Write your answer on the line.
What is the exact length of a side of the square? _________________ inches
Answer:
50 or 5 2
12
Sample item for performance expectation G.3.B/M2.3.F
In the diagram:
 Quadrilateral ABCD with AB  DC
 ABD  CDB
Prove ABD  CDB using mathematical language and concepts.
Proof:
2-point response: The student shows understanding of proving triangle congruence by doing the
following:



Writes AB  DC and ABD  CDB (given)
Writes BD  BD by reflexive property, or equivalent
Writes ABD  CDB by Side-Angle-Side, or equivalent
1-point response: The student does one of the following:
 Writes BD  BD by reflexive property, or equivalent

Writes BD  BD and writes ABD  CDB by Side-Angle-Side, or equivalent
Note: Student responses may be in the form of a flow chart proof, two-column proof, or paragraph proof.
Student responses should refer to the given information within the proof.
13
Sample item for Performance Expectation G.7.B (G.3.D)/M2.6.B (M2.3.G)
The length of one diagonal of a rectangle is 17 feet. The length of the rectangle is
7 feet greater than its width.
Determine the perimeter of the rectangle.
Show your work using words and/or numbers.
What is the perimeter of the rectangle? _______________ feet
2-point response: The student shows understanding of solving a problem by doing the following:
Understanding:
 Uses 17 and 7
Strategy:
 Shows a strategy to determine the perimeter of the rectangle
Answer:
 Writes 46.
1-point response: The student does one of the following:
 Uses 17 and 7 and shows a strategy that could lead to the perimeter of the rectangle
 Writes 46.
14
Sample item for Performance Expectation G.3.E/M2.3.H
Two trees stand on opposite banks of the river, located at points A and B. A surveyor is
standing at point C which is 100 feet from point B. He measures angle ACB to be 72
degrees.
Determine the length of AB, the width of the river to the nearest hundredth.
Show your work using words and/or numbers.
What is the length of AB to the nearest hundredth? _______________ feet
2-point response: The student shows understanding of solving problems involving basic trigonometric
ratios by doing the following:
 Shows use of tangent such as
, or equivalent.

Writes 307.77 feet.
1-point response: The student does one of the following:
 Shows use of tangent such as
, or equivalent.

Writes 307.77 feet.
15
Sample item for Performance Expectation G.3.G/M2.3.K
The measure of an exterior angle of a regular polygon is 40 degrees.
Determine the number of sides the regular polygon has.
Write your answer on the line.
How many sides does the regular polygon have? _________________
Answer: 9
Sample item for Performance Expectation G.3.H
In circle A,
.
Determine the measure of
.
Write your answer on the line.
What is the measure of
? _________________ (degrees)
Answer: 67.5
16
Sample Question G3B1:
STUDY NOTES:
Note: the vertical angles would ordinarily be written as angle ABD is congruent to EBC, but since the
angle is the same as ABD is congruent to CBE, either answer would still score as a 2-point response.
17
Sample item for Performance Expectation G.3.B/M2.3.F
The triangles shown are congruent.
Which postulate or theorem could be used to conclude that the triangles are congruent?
O A.
AAS
O B.
ASA
O C.
SAS
O D.
SSS
Answer: A
18
Sample Question G3D1:
STUDY NOTES:
For this problem, you do the Pythagorean Theorem two times. First, you find the diagonal (hypotenuse)
for the base of the box. After you have calculated this, then use this leg as the base for the second right
triangle that has the height as the second leg. Calculate this diagonal (hypotenuse) for length AB.
19
Sample Question G3G1:
Answer: D
STUDY NOTES:
The midsegment of a trapezoid is the average of the bases.
20
Sample item for Performance Expectation G.7.B (G.3.C,E)/M2.6.B (M2.3.I,H)
A lighthouse stands on a hill 80 meters above sea level. The measure of
and the measure of
is 30 degrees.
Drawing is not to scale.
What is the height of the lighthouse?
O A.
80 meters
O B.
120 meters
O C.
160 meters
O D.
240 meters
Answer: C
21
is 60 degrees
Sample item for Performance Expectation G.3.G/ M2.3.K
Quadrilateral LMNO is a parallelogram.
L
M
N
O
Which statement about the parallelogram must be true?
O A.
O B.
O C.
O D.
is the bisector of
Diagonals
and
are perpendicular.
Answer: B
Sample item for Performance Expectation G.4.A/M1.3.H
Line w passes through the point (5, 9) and is parallel to the line y = 3.
Determine an equation that represents line w.
Write your answer on the line.
What is an equation that represents line w ? _________________
Answer: y = 9
22
Sample item for Performance Expectation G.4.B/M2.3.L
Points R, S and T are collinear. S is the midpoint of
.
The coordinates of point R are (-4, 5). The coordinates of point S are (-1, 3).
Determine the coordinates of point T.
Write your answer on the line.
What are the coordinates of point T ? (
Answer: (2, 1)
23
,
)
Sample Question G4B1:
Answer: (1,-2)
24
Sample Question G4B2:
25
Sample Question G4C6:
Below is a diagram of a miniature golf hole as drawn on a coordinate grid. The dimensions of
the miniature golf hole are 4 feet by 12 feet and the space is enclosed with a wall so that players can
bounce their golf shots off of the four surrounding walls. There are three tees to choose from: Player 1
must start his ball from (1, 3). The actual hole is located at (10, 3). A wall separates the tees from the
hole. Sketch the intended path of the ball so that the player banks off of the lower wall. If Player 1 hits a
hole in one, what is the shortest distance that the ball traveled?
A. 145 feet
B. √117 feet
C. 13 feet
D. √145 feet
Answer: B
STUDY NOTES:
You can reflect an imaginary construction line from Player 1’s starting point at (1, 3) over the x-axis to an
imaginary point at (1, -3). Drawing a line from this point to the hole at (10,3) is the total distance. This
imaginary path creates congruent angles for the bank shot from the tee to the wall, and from the wall
up to the hole. Use Pythagorean Theorem or distance formula.
92 + 62 = √117 or use the distance formula below:
26
Sample item for Performance Expectation G.5.A
Triangle RST is shown on the coordinate grid. Point R is located at (0, -2).
y
x
R
T
S
Triangle RST is first reflected across the x-axis and then reflected across the line y = x to
create triangle R”S”T”.
What are the coordinates of the vertex T” of triangle R”S”T” ?
O A.
(7, 3)
O B.
(7, -3)
O C.
(3, 7)
O D.
(3, -7)
Answer: A
27
Sample item for Performance Expectation G.6.C
The figure shown is a triangular pyramid. The height of the pyramid is 8 meters and the
base is an equilateral triangle with a base edge length of 6 meters.
Determine the volume of the triangular pyramid.
Write your answer on the line.
What is the volume of the triangular pyramid? _________________ cubic meters
Answer:
or approximately 41.57
Sample Question G6F1:
28
Sample Question G6F2:
Sample Question G6F3:
Answer: A
STUDY NOTES:
To convert from cubic yards to cubic feet, you multiply by 3 feet for each of the dimensions, so: 33 or 27.
29
Sample Question G6F4:
Sample Question G6F5:
30
Sample Question G6F6:
31