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Quizzes > High School Quizzes > Mathematics

Master Proportions in Triangles Practice Quiz

Boost Your Confidence with Triangle Similarity Proofs

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

In similar triangles, which of the following is true about corresponding angles?
They are complementary
They are equal
They are supplementary
They add up to 180°
In similar triangles, corresponding angles are congruent. This means each angle in one triangle is equal to its corresponding angle in the other triangle, ensuring the triangles have the same shape.
What does the term 'scale factor' refer to in the context of similar triangles?
The difference between the longest and shortest sides
The ratio of the perimeters of two triangles
The factor by which corresponding sides are multiplied
The sum of the angles in a triangle
The scale factor indicates how much a triangle is enlarged or reduced relative to its similar counterpart. It is calculated by dividing a side length in one triangle by the corresponding side length in the other.
Given two similar triangles, if a side in the first triangle is 4 and its corresponding side in the second triangle is 6, what is the scale factor from the first triangle to the second?
4
2
1.5
0.67
The scale factor is obtained by dividing the corresponding side of the second triangle by that of the first (6/4 = 1.5). This factor shows how much larger one triangle is compared to the other.
Which criterion is NOT valid for proving triangle similarity?
HL (Hypotenuse-Leg)
SAS (Side-Angle-Side)
SSS (Side-Side-Side)
AA (Angle-Angle)
HL (Hypotenuse-Leg) is used to prove congruence in right triangles rather than similarity. The valid criteria for triangle similarity include AA, SSS, and SAS.
If a triangle has angles of 30°, 60°, and 90°, what are the angles of any triangle similar to it?
They can be different as long as the side ratios match
They are 30°, 60°, and 90°
They sum to 180° but are otherwise unknown
They are all 60°
Similar triangles have the same angle measures, so any triangle similar to one with angles of 30°, 60°, and 90° will also have those exact angles. This is a direct result of the Angle-Angle similarity postulate.
Triangles MNO and XYZ are similar. If the scale factor from triangle MNO to triangle XYZ is 3 and side MN measures 4, what is the length of side XY?
12
7
8
3
The length of a corresponding side is found by multiplying the side by the scale factor. Here, 4 multiplied by 3 gives 12.
Triangle ABC is similar to triangle DEF. If triangle ABC has side lengths 3, 4, and 5, and the side corresponding to 3 in triangle DEF is 6, what are the side lengths of triangle DEF?
6, 8, and 10
6, 7, and 10
6, 8, and 9
5, 8, and 10
The scale factor is 6/3 = 2, which when applied to the other sides (4 and 5) gives 8 and 10 respectively. This maintains the proportionality of corresponding sides in similar triangles.
Triangles ABC and DEF are similar with sides AB = 2x, BC = 3x, and DE = 10, where AB corresponds to DE. What is the value of x?
4
6
5
2
Setting up the proportion for corresponding sides (2x = 10) gives x = 5. This is a common approach for solving unknowns in similar triangles.
Which of the following statements is false regarding ratios in similar triangles?
The ratios of corresponding sides are equal
The ratio of corresponding sides is equal to the ratio of the areas
The ratio is the same as the ratio of the perimeters
The ratios can be used to solve for unknown side lengths
The ratio of areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, stating that the ratios are equal is false.
If two similar triangles have areas of 25 and 100 respectively, what is the scale factor between their corresponding sides from the smaller triangle to the larger triangle?
2
4
10
5
The scale factor is the square root of the ratio of the areas, so √(100/25) = √4 = 2. This means each side of the larger triangle is twice as long as the corresponding side of the smaller triangle.
If in similar triangles the ratio AB/DE equals AC/DF, what does this imply about triangles ABC and DEF?
They have the same area
They have identical side lengths
They are congruent
They are similar
The equality of the ratios of corresponding sides confirms triangle similarity. Although the triangles share the same shape, their sizes can differ.
In similar triangles ABC and PQR, if the sides of ABC are 3, 5, and 7, and the corresponding sides of PQR are 6, 10, and x, what is the value of x?
14
15
12
13
The scale factor from triangle ABC to PQR is 6/3 = 2. Multiplying the third side (7) by 2 gives x = 14, maintaining the proportionality of the triangles.
Which method is most effective for finding an unknown side in similar triangles?
Subtracting non-corresponding sides
Cross multiplying proportions
Using the Pythagorean theorem
Adding corresponding sides
Cross multiplying the proportions established by corresponding sides is the most direct method for solving unknown side lengths in similar triangles.
In a pair of similar triangles, if the ratio of their corresponding sides is 3:4 and the smaller triangle has a perimeter of 18, what is the perimeter of the larger triangle?
26
20
24
22
The perimeter of the larger triangle is determined by multiplying the smaller triangle's perimeter by the scale factor (4/3). Thus, 18 × (4/3) = 24.
Two similar triangles have corresponding altitudes in the ratio 2:3. What is the ratio of their areas?
3:2
9:4
4:9
2:3
The areas of similar triangles are proportional to the square of the ratio of their corresponding sides. Hence, (2/3)² = 4/9, which is the area ratio.
In similar triangles, all corresponding linear measurements are scaled by the same factor. If the median corresponding to side AB in triangle ABC is 8 and the scale factor from triangle ABC to triangle DEF is 1:1.5, what is the length of the corresponding median in triangle DEF?
12
14
16
10
In similar triangles, every linear measurement, including medians, is scaled by the same factor. Multiplying 8 by 1.5 gives 12.
In a right triangle, the altitude to the hypotenuse divides it into two segments. According to the geometric mean theorem, if the altitude is 5, what is the product of the lengths of these two segments?
25
30
20
10
The geometric mean theorem states that the square of the altitude equals the product of the segments it creates on the hypotenuse. Since 5² = 25, the product is 25.
Triangles X and Y are similar. If the ratio of corresponding sides from triangle X to triangle Y is 2:5 and the circumradius of triangle Y is 15, what is the circumradius of triangle X?
7
5
6
8
All linear dimensions in similar triangles, including the circumradius, scale by the same factor. Multiplying 15 by 2/5 gives 6.
In triangle ABC, a line drawn parallel to side BC intersects sides AB and AC, dividing them in a ratio of 3:4. What is the ratio of the area of the smaller triangle (formed by the parallel line) to the area of triangle ABC?
3:4
16:9
4:3
9:16
When a line parallel to one side divides the other two sides in a ratio, the area ratio of the smaller to the larger triangle is the square of that ratio. Here, (3/4)² = 9/16.
Using similar triangles, if a 1-meter stick casts a 0.5-meter shadow and a tree casts a 10-meter shadow, how tall is the tree?
20 m
15 m
5 m
10 m
Since the triangles formed by the stick and the tree with their respective shadows are similar, the ratio of height to shadow remains constant. The tree's height is calculated as 10 / 0.5 = 20 m.
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Study Outcomes

  1. Understand the criteria for triangle similarity, including AA, SAS, and SSS.
  2. Identify similar triangles through angle and side comparisons.
  3. Apply proportional reasoning to compute missing side lengths in similar triangles.
  4. Analyze relationships between corresponding sides and angles to solve geometry problems.
  5. Evaluate test-ready skills by solving practice questions related to triangle proportions.

Proportions in Triangles Cheat Sheet

  1. Definition of Similar Triangles - Triangles are similar when their corresponding angles match and their sides stay in the same proportion, like mini-me versions of each other. Grasping this definition is your golden ticket to unlocking many geometry puzzles with ease. byjus.com
  2. Angle-Angle (AA) Criterion - If two angles of one triangle are equal to two angles of another, the triangles are hands-down similar. It's like finding a secret handshake between shapes that instantly tells you they belong together. byjus.com
  3. Side-Angle-Side (SAS) Criterion - When a pair of sides around an angle are proportional and the included angle is equal, the triangles must be similar. Think of it as having the same recipe: matching key ingredients around the main flavor point ensures identical taste. byjus.com
  4. Side-Side-Side (SSS) Criterion - If all three sides of one triangle are in proportion to the three sides of another, they're similar without a doubt. It's like using the same zoom setting on two photos - they look identical, just different sizes. byjus.com
  5. Properties of Similar Triangles - Corresponding angles are equal and corresponding sides are proportional. These rules let you swap, stretch, and compare triangles like building blocks in your geometry toolkit. byjus.com
  6. Basic Proportionality Theorem (Thales' Theorem) - A line dividing two sides of a triangle proportionally will be parallel to the third side. This nifty trick helps you break down complex figures into simpler, solvable pieces. byjus.com
  7. Area Relationship - The ratio of areas of similar triangles equals the square of the ratio of their corresponding sides. Watching shapes grow (or shrink) and seeing their areas change by a squared factor is pure math magic. byjus.com
  8. Practice Identifying Similar Triangles - Hunt for similar triangles in stars, architecture, or doodles in your notebook to sharpen your skills. The more you spot, the faster you'll ace those similarity questions! GeeksforGeeks
  9. Real-World Applications - Use triangle similarity to measure heights with shadows, like finding a tree's height by comparing it to a stick's shadow. It's practical geometry that turns you into a measurement ninja outdoors! GeeksforGeeks
  10. Equilateral Triangles Are Always Similar - Every equilateral triangle has three equal angles and sides, so they're all just resized versions of one another. Remember this handy shortcut whenever you spot an equilateral shape! GeeksforGeeks
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