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Quizzes > High School Quizzes > Mathematics

Similarity Unit Test Practice Quiz

Master similarity concepts with practical study questions

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art depicting a trivia quiz on geometric similarity for high school students.

What does it mean when two figures are similar?
They have the same shape but not necessarily the same size.
They have the same size but not the same shape.
They are congruent and have identical sizes.
They have different angles but proportional sides.
Similar figures share the same shape even if their sizes differ. The corresponding angles remain equal, and the sides are in proportion.
In similar triangles, which of the following is always true?
Corresponding angles are equal.
All corresponding sides are equal.
The triangles have the same area.
The triangles have the same perimeter.
Similar triangles have their corresponding angles equal, while the sides are proportional rather than equal. This is a core property defining similarity.
If two triangles are similar, what can be said about the ratio of any two corresponding sides?
The ratio is constant for all pairs.
The ratio changes depending on the side.
The ratio is always 1.
The ratio equals the sum of the sides.
For similar triangles, every pair of corresponding side lengths has the same constant ratio. This constancy is what allows us to apply scaling uniformly.
Which similarity criterion is most commonly used to prove that two triangles are similar?
AA (Angle-Angle).
SSS (Side-Side-Side).
SAS (Side-Angle-Side).
HL (Hypotenuse-Leg).
The AA criterion is widely used because if two angles of one triangle are equal to two angles of another, the triangles are similar. This method simplifies the process of proving similarity.
A dilation in geometry is used to transform a figure into a figure that is:
Similar to the original figure.
Congruent to the original.
Reflected across a line.
Rotated around a point.
Dilation scales a figure up or down while preserving its shape, which means the new figure is similar to the original. This transformation changes size but not the overall geometric proportions.
Triangle ABC is similar to triangle DEF with a scale factor of 3. If side AB = 5, what is the length of the corresponding side DE?
15
8
5
10
A scale factor of 3 means every side in triangle ABC is tripled to produce the corresponding side in triangle DEF. Hence, multiplying 5 by 3 gives 15.
In two similar rectangles, the ratio of the lengths of their sides is 2:5. If the smaller rectangle's width is 4, what is the width of the larger rectangle?
10
8
9
5
The ratio 2:5 indicates that for every 2 units in the smaller rectangle, the larger one has 5 units. Setting up the proportion with 4 yields the larger width of 10.
If two similar triangles have areas in the ratio of 9:16, what is the ratio of their corresponding side lengths?
3:4
9:16
1:2
2:3
The area ratio of similar figures is the square of the ratio of their corresponding side lengths. Taking the square root of 9:16 results in a side ratio of 3:4.
In triangle ABC, angle A is 30° and angle B is 60°. If triangle ABC is similar to triangle DEF, what must be the measure of angle D corresponding to angle A?
30°
60°
90°
120°
Since corresponding angles in similar triangles are equal, angle D must be 30° to match angle A of triangle ABC. This is a fundamental property of similar triangles.
Two regular pentagons are similar, and the perimeter of the smaller pentagon is 20. If the similarity ratio is 1:3, what is the perimeter of the larger pentagon?
60
40
30
50
The perimeter scales directly with the similarity ratio. Multiplying the smaller perimeter of 20 by 3 yields a larger perimeter of 60.
In similar polygons, if the scale factor is k, what is the ratio of their areas?
k^2
k
2k
k^3
The areas of similar figures change by the square of the scale factor. Therefore, if the scale factor is k, the ratio of the areas is k^2.
For two similar triangles, if one triangle has sides 3, 4, 5, and the scale factor is 2, what are the lengths of the sides of the similar triangle?
6, 8, 10
5, 7, 9
6, 7, 8
3, 4, 10
Multiplying each side of the triangle by the scale factor of 2 gives the sides of the similar triangle. That is, 3 becomes 6, 4 becomes 8, and 5 becomes 10.
In two similar triangles, if the ratio of the perimeters is 1:4, which of the following is true about the ratio of their areas?
1:16
1:4
1:8
1:2
Since the area of similar figures scales as the square of their linear dimensions, a perimeter ratio of 1:4 leads to an area ratio of 1:16. This underscores the quadratic relationship between side lengths and area.
Given two similar triangles, triangle XYZ and triangle PQR, if XY = 8 and PQ = 12, what is the similarity ratio?
2:3
3:2
4:5
5:4
Dividing the side lengths 8 and 12 gives 8/12, which simplifies to 2:3. This ratio represents the constant factor by which the triangles differ in size.
A line parallel to one side of a triangle divides the other two sides proportionally. What theorem does this statement represent?
Basic Proportionality Theorem
Pythagorean Theorem
Triangle Sum Theorem
Angle Bisector Theorem
This property is known as the Basic Proportionality Theorem, which states that a line parallel to one side of a triangle will cut the other two sides in the same ratio. It is a fundamental concept in proving triangle similarity.
Triangle ABC is similar to triangle DEF with a scale factor of 5. If the area of triangle ABC is 8, what is the area of triangle DEF?
200
40
8
125
Since the area scales by the square of the scale factor, a factor of 5 means the area becomes 5² times larger. Multiplying 8 by 25 gives 200.
In two similar trapezoids, the length of the smaller trapezoid's longer base is 10 and the similarity scale factor is 1:1.5, what is the length of the larger trapezoid's longer base?
15
13
20
12
Multiplying the smaller trapezoid's base of 10 by the scale factor 1.5 results in 15. This direct multiplication demonstrates the linear scaling in similar figures.
A triangle's sides are in the ratio 3:4:5. If scaling the triangle by a factor of k results in an area of 150, what is a possible value for k?
5
3
10
15
A 3-4-5 triangle has an area of 6. When scaled by a factor k, the area becomes 6k². Setting 6k² equal to 150 gives k² = 25, so k = 5.
In similar solids, the ratio of their surface areas is the square of the scale factor, while the volume ratio is the cube. How would the volume of a solid change if its surface area is quadrupled?
The volume increases by a factor of 8.
The volume doubles.
The volume increases by a factor of 4.
The volume remains the same.
Quadrupling the surface area means the square of the scale factor is 4, so the scale factor is 2. Since volume scales by the cube of the scale factor, the volume increases by 2³, which is 8.
A triangle is inscribed in a circle, and a line through one vertex is drawn parallel to the opposite side, creating a smaller, similar triangle within the larger triangle. If the ratio of the corresponding side lengths of the small triangle to the large triangle is r, what is the ratio of their areas?
r^2
r
r^3
2r
For similar triangles, the ratio of the areas is the square of the ratio of their corresponding side lengths. Thus, if the side ratio is r, the area ratio is r².
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Study Outcomes

  1. Analyze geometric figures to identify and justify similarity using corresponding angles and proportional sides.
  2. Apply the criteria for triangle similarity to determine relationships between different geometric shapes.
  3. Solve problems involving the calculation of scale factors and the application of proportional reasoning in similar figures.
  4. Interpret exam-style questions and diagrams to develop effective strategies for addressing similarity challenges.

Similarity Unit Test Part 1 Cheat Sheet

  1. Understanding Similar Figures - Two figures are called similar when they have the exact same shape, even if one is a miniature model of the other. This means every angle matches up, and corresponding sides keep a constant ratio. Once you see shapes playing matchmaker, similarity feels like a piece of cake! Read more about Similar Figures
    2. cuemath.com
  2. Similarity Transformations - Think of dilations, rotations, reflections and translations as your shape's makeover squad - they change size or orientation but never its style. Mastering these moves helps you map one figure onto another and confirm they're cut from the same geometric cloth. It's like learning the secret handshake of shapes! Explore Transformations
    3. geometrycommoncore.com
  3. Angle-Angle (AA) Similarity Criterion - If two angles of one triangle line up perfectly with two angles of another triangle, the whole triangles must be similar. Since the sum of angles in a triangle is always 180°, matching two guarantees the third falls into place like a puzzle piece. It's the quickest shortcut to spotting twin triangles! Dive into AA Criterion
    4. learner.org
  4. Side-Angle-Side (SAS) Similarity Criterion - When one angle of a triangle matches another and the sides around those angles are in proportion, you've unlocked SAS similarity. This combo of angle harmony and side ratio checks is like using two-factor authentication for triangle twins. It keeps your proofs both tight and satisfying! Check out SAS Criterion
    5. learner.org
  5. Side-Side-Side (SSS) Similarity Criterion - If all three pairs of corresponding sides of two triangles share the same ratio, the triangles must be similar - no angles needed. Comparing side lengths is like measuring three sides of a secret code; once they match, the similarity is confirmed. It's the "three strikes, you're in" rule for triangles! Learn about SSS Criterion
    6. learner.org
  6. Proportionality Theorems - The Basic Proportionality Theorem (Thales' Theorem) says a line parallel to one side of a triangle splits the other sides in matching ratios. Meanwhile, the Angle Bisector Theorem divides the opposite side into segments proportional to the adjacent sides. Together, these theorems are your go‑to tools for slicing and dicing triangles with confidence! Explore Proportionality
    7. collegesidekick.com
  7. Perimeters of Similar Figures - When figures are similar, the ratio of their perimeters equals the ratio of their corresponding side lengths. So if one triangle is twice as big on each side, its perimeter is also twice as large. It's the straight‑forward "scale factor" rule that keeps everything in proportion! Discover Perimeter Ratios
    8. byjus.com
  8. Areas of Similar Figures - The area ratio of two similar figures is the square of their side‑length ratio. That means a figure twice as big on each side will cover four times the area! Remembering this squared relationship is a fast pass to area calculations. Unpack Area Ratios
    9. mathnirvana.com
  9. Applications of Similarity - Similarity shows up everywhere: mapmakers use it to shrink real distances, photographers rely on it to capture scenes, and architects sketch scale models to plan buildings. Spotting similarity in the real world helps you connect classroom theory to everyday magic. It's geometry's way of making life more relatable! See Real‑World Uses
    10. schooltube.com
  10. Practice Problems - The secret sauce to mastering similarity is practice: determine triangle similarity by matching angles or sides, solve for missing lengths with proportions, and tackle dilation scenarios with scale factors. Each problem you conquer builds confidence and cements these concepts for good. Grab your calculator and start cracking codes! Get Practice Problems
    11. collegesidekick.com
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