Geometry Quarter Exam - Geometry Problems, Solutions, and Explanations

This document presents a complete solution to a Geometry Quarter Exam, covering a wide array of geometric concepts. The exam includes problems on solving proportions, identifying similar triangles, determining triangle types (acute, right, obtuse), and finding missing values in right triangles. It also addresses transformations (reflection, translation, rotation), finding areas of complex figures, calculating arc measures, and determining the area and surface area of various 2D and 3D shapes (octagons, cylinders, rectangular prisms, cones, spheres). Additionally, the solution provides equations of circles, solves for unknown variables in geometric figures, and covers probability calculations using spinners and number cubes, alongside permutations and combinations.

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
1. Solve the proportion. Show your work. 3
x+1 = 5
11
Solution:
Simplify further
Hence,
2. The figures in each pair are similar. Find the value of each variable. Show your work.
a.
Solution:
b.

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
Solution:
3. Determine whether the triangles are similar. If so, write a similarity statement and name
the postulate or the theorem you used. If not, explain.
a.
Solution: Consider the triangles ABC and PQR.
Since,
Then by Angle-Angle-Angle property, triangles ABC and PQR are similar.

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
b.
Solution: Consider triangle BAC, since this implies that triangle BAC is a
right angle triangle.
Consider triangle EDF, since this implies that triangle EDF is a right angle
triangle. Since both triangles are right angle but their corresponding sides are not congruent
(equal). Hence, triangles are not similar.
4. Find the value of each variable for the right triangles. Make sure all answers are in
reduced radical form. Show your work.
a.
Solution: Since given triangle is right angle, use Pythagorean
Theorem we get

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
(b)
Solution:
Since given triangle is right angle, use Pythagorean theorem we get

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
5. Given the following triangle side lengths, identify the triangle as acute, right or obtuse.
Show your work.
a. 5in, 6in, 7in
Solution: Suppose that are the
length of sides of triangle where side is
largest. Then triangle is
Right angle if
Acute angle if
Obtuse angle if
Since, this implies that
Triangle is Acute.
b) 18in, 9in, 12in
Solution: Since this implies that
Triangle is Obtuse.
6. Find the missing value. Show your work. Round to the nearest hundredth.

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
a.
Solution:
b.
Solution:
7. Given the vertices of ∆ABC are A (2,-5), B (-4,6) and C (3,1), find the vertices following
each of the transformations FROM THE ORIGINAL vertices:
a. Rx = 3 b. T<3,-6>

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
c. r(90◦, o)
Solution a: R (x-axis) is the reflection of the original triangle ABC on the x-axis. The new
coordinates are given as D (2, 5), E (-4, -6), F (3, -1).
Solution b: T (3, -6) is the translation of the original triangle ABC three units right and six units
down. The new coordinates would be A'(5, -11), B'(-1, 0), and C'(6, -5).
Solution c: r (90°, 0) is the rotation of the original triangle ABC on the origin by 90° clockwise.
The new coordinates would be A'(-5, -2), B'(6, -4) and C'(1 -3).
8. Find the congruence transformation that maps ∆ABC to ∆A’B’C’. Explain.
Solution:
9. Find the area of each figure to the nearest tenth.

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
a.
Solution a:
The area of the given figure = area of rectangle ABEF + area of triangle CDE
Area of rectangle ABEF =
Let’s find area of triangle DEF
Since ED = EF – AB = 180 – 110 = 70 cm and EC = EB – BC = 140 – 50 = 90 cm
So area of triangle DEF
Hence, required area of figure =

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
b.
Solution b: Area of quadrilateral ABCED = area of square ABCE + area of triangle AED
Let’s first find area of square ABCE. Since AB = 8 ft and AE = 8 ft
So area of square ABCE
Let’s first find area of triangle AED. Since AED is right angle triangle
So,
So area of triangle AED
Hence, required area = 64 + 24
Find the measure of both arc AB and arc APB, where <APB = 108◦.Please be sure to show your
steps.

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
Solution:
10. Find the area of a regular Octagon with side lengths 5in. Round your answer to the
nearest tenth.
Solution: The area of a regular octagon of side length a is given by
Given that
So,
11. Find the surface area of each figure to the nearest tenth.
a. b.

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
Solution a: Surface Area of a Cylinder
Given that
So,
Surface Area of a Cylinder
Solution b: The formula for surface area of rectangular prism is
Given that
So, surface area
b) What is a real-world example or use of surface area?
12. Find the surface area of each figure to the nearest tenth.
a. b.
Solution a:
The surface are of given figure is

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
Solution b: Given that the height and radius of the cone are 8 in and 3 in respectively.
The slant height
So the surface area of the cylinder is
13. Find the volume of each figure to the nearest tenth.
a.
b.
c.
Solution a: Given that the radius of sphere is . The volume of the sphere is

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
Solution b: Given that the radius of cone is and slant height . Since height
The volume of the sphere is
Solution c: Given that the length, breath and height of the cuboids are
The volume of cuboids is
14. Find the center and radius, then write the standard equation of each circle
a.
b.

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
c.
Solution a: From the graph, the center and radius of the circle are and 5 respectively. The
standard form of equation is
Solution b: From the graph, the center and radius of the circle are and 3 respectively. The
standard form of equation is
Solution c: From the graph, the center and radius of the circle are and 5 respectively. The
standard form of equation is

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
15. Solve for x
a.
b.
Solution a: We know that if two tangents intersect outside of a circle, the distances along the
tangent from the intersection to the circumference of the circle are equal.
So,
Solution b:
16. Evaluate

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
a. 6! b. 8P5 c. 12C4
Solution a: Since,
So,
Solution b: Since,
So,
Solution c: Since,
So,
17. Use the spinner to find each theoretical probability

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
a. P (a number no more than 5)
b. P (an even number)
c. P (a number less than 3)
Solution a: There are four numbers 1, 2, 3, and 4 which are less than 5. Hence,
P (a number no more than 5)
Solution b: There are four even numbers 2, 4, 6 and 4. Hence,
P (an even number)
Solution c: There are two numbers 1 and 4 which are less than 3. Hence,
P (a number less than 3)
18. You roll two standard number cubes. What is the probability that the sum is odd, given
than one of the number cubes shows a 4?
Solution: It is given that one of the number cubes shows a 4, and if the sum is odd, then
the second cubes must shows 1 or 3, or 5. That is the number of favorable cases = 3 and

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
number of possible cases = 6
hence, the probability that the sum is odd
19. What is the area of the figure:
Solution: Since triangle ACB is right angle triangle.
So,
And

Geometry Quarter Exam
Name: ____________________________ Student Number: ____________________
Answer the questions below. Make sure to show your work and provide complete geometric explanations for full
credit.
So, Area of triangle BCA