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Quizzes > High School Quizzes > Mathematics

Right Triangle Similarity Practice Quiz

Ace Your Exam with Quick Check Tips

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Colorful paper art promoting a fast-paced geometry trivia quiz for high school students.

In similar right triangles, which of the following properties is always true?
Their corresponding angles are congruent and corresponding sides are proportional.
They have equal side lengths.
They have identical areas.
They always have the same hypotenuse length.
Similar right triangles are defined by having congruent corresponding angles and proportional corresponding side lengths. This fundamental property distinguishes similarity from congruence.
What is the common ratio in similar right triangles?
The ratio of the areas of the triangles.
The ratio of the angles of the triangles.
The ratio of the lengths of any pair of corresponding sides.
The ratio of the perimeters of the triangles.
The constant ratio between the lengths of corresponding sides in similar triangles is the defining characteristic of similarity. This means any pair of corresponding sides will give the same ratio.
Which side in a right triangle is referred to as the hypotenuse?
Any leg of the triangle.
The shortest side of the triangle.
The median drawn to the right angle.
The side opposite the right angle.
By definition, a right triangle's hypotenuse is the side opposite the right angle and is its longest side. This is a key distinction in right triangle geometry.
In right triangles, if one acute angle of one triangle is equal to an acute angle of another, what can be concluded?
The triangles have the same side lengths.
The triangles have equal areas.
The triangles are congruent.
The triangles are similar.
When two right triangles share one acute angle, they each already have a right angle which guarantees two angles are equal. This satisfies the AA (Angle-Angle) criterion for similarity.
Which characteristic distinguishes the hypotenuse from the legs in a right triangle?
It is the side adjacent to the right angle.
It is always equal in length to one of the legs.
It is the side opposite the right angle and the longest side.
It bisects the right angle.
In any right triangle, the hypotenuse is uniquely identified as the side opposite the right angle, and it is always the longest side. This property is essential in many aspects of right triangle geometry.
In two similar right triangles, if the hypotenuse of the smaller triangle is 5 cm and the corresponding hypotenuse of the larger triangle is 10 cm, what is the scale factor from the smaller to the larger triangle?
10
5
2
0.5
The scale factor is determined by dividing the corresponding side length of the larger triangle by that of the smaller triangle. Here, dividing 10 cm by 5 cm gives a scale factor of 2.
In a right triangle, drawing an altitude to the hypotenuse creates two smaller right triangles. Which of the following relationships is true regarding the altitude?
The altitude is equal to the sum of the segments on the hypotenuse.
The altitude is the average of the two legs.
The altitude is proportional to the difference of the hypotenuse segments.
The square of the altitude equals the product of the lengths of the two segments it creates on the hypotenuse.
This is a statement of the geometric mean theorem, which applies when an altitude is drawn to the hypotenuse in a right triangle. The theorem establishes that the altitude squared equals the product of the two hypotenuse segments.
Given two similar right triangles, if the smaller triangle has legs of lengths 3 and 4, and the corresponding shorter leg of the larger triangle is 6, what is the length of the larger triangle's leg corresponding to the 4?
8
10
7
6
The scale factor from the smaller to the larger triangle is 6 divided by 3, which equals 2. Multiplying the other leg (4) by 2 gives 8.
If triangle ABC is similar to triangle DEF and triangle ABC has side lengths 6, 8, and 10, what are the corresponding side lengths of triangle DEF if the similarity scale factor is 3?
15, 20, 25
9, 12, 15
18, 24, 30
12, 16, 20
Multiplying each side of triangle ABC by the scale factor 3 yields the side lengths of the similar triangle: 6Ã - 3 = 18, 8Ã - 3 = 24, and 10Ã - 3 = 30. This preserves the ratio of side lengths.
In similar right triangles, how is the ratio of their areas related to the similarity scale factor?
The ratio of the areas is equal to the square of the scale factor.
The ratio of the areas is equal to the scale factor itself.
The ratio of the areas is equal to the cube of the scale factor.
The areas remain the same regardless of the scale factor.
When figures are similar, their areas scale by the square of the scale factor used for the side lengths. This quadratic relationship is a fundamental concept in similarity.
Which result in a right triangle is always true when you multiply the two legs together and then divide by the hypotenuse?
It provides the median length to the hypotenuse.
It gives the sum of the legs.
It equals the difference between the legs.
It equals the length of the altitude drawn to the hypotenuse.
A standard property of right triangles is that the product of the lengths of the legs divided by the hypotenuse equals the length of the altitude to the hypotenuse. This comes from the relationships in similar triangles formed by the altitude.
Which theorem is used to establish the similarity of the two smaller right triangles formed by the altitude to the hypotenuse in a right triangle?
SAS Similarity Criterion
Pythagorean Theorem
AA (Angle-Angle) Similarity Postulate.
HL Congruence Theorem
The AA Similarity Postulate states that two triangles are similar if they have two corresponding angles that are equal. In a right triangle with an altitude, both of the smaller triangles share the right angle and one other common angle with the original triangle.
In right triangle similarity, if the ratio of the legs in one triangle is 3:4, what will be the ratio of the legs in any triangle similar to it?
4:3
3:4
1:2
3:5
Similar triangles maintain the same proportional relationships among corresponding sides. Therefore, a triangle similar to one with legs in a 3:4 ratio will also have its legs in a 3:4 ratio.
Which of the following is a sufficient condition to prove two right triangles are similar?
All corresponding sides are of equal length.
Their hypotenuses are equal in length.
One acute angle in one triangle is congruent to one acute angle in the other.
Their areas are identical.
For right triangles, knowing that one of the acute angles is equal between them is enough to conclude similarity by the AA criterion. The right angle is common, so only one additional pair of equal angles is needed.
Given a 6-8-10 right triangle similar to another triangle with a hypotenuse of 25, what is the similarity scale factor?
2.5
1.5
2
3
The scale factor is calculated by dividing the larger triangle's hypotenuse by the smaller triangle's hypotenuse. Dividing 25 by 10 yields a scale factor of 2.5.
In a right triangle with an altitude drawn to the hypotenuse, if the hypotenuse is divided into segments of lengths 9 and 16, what is the length of the altitude?
13
10
15
12
According to the geometric mean theorem, the square of the altitude drawn to the hypotenuse equals the product of the two segments into which it divides the hypotenuse. Here, 9 multiplied by 16 is 144, and the square root of 144 is 12.
If a small right triangle with legs of 5 and 12 (and hypotenuse 13) is similar to a larger right triangle with hypotenuse 39, what is the length of the leg in the larger triangle corresponding to the leg of 12 in the smaller triangle?
36
26
30
40
The smaller triangle is a 5-12-13 triangle, so the scale factor from the smaller to the larger triangle is 39/13, which equals 3. Multiplying 12 by 3 gives 36 as the corresponding leg length in the larger triangle.
When two similar right triangles have a corresponding side ratio of 3:5, what is the ratio of their perimeters?
3:5
3:4
5:3
9:25
Since perimeters scale linearly with the side lengths of similar triangles, the ratio of the perimeters is the same as the ratio of their corresponding sides. Therefore, the ratio remains 3:5.
Which formula correctly expresses the geometric mean relationship involving the altitude of a right triangle drawn to the hypotenuse?
Altitude = (Segment 1 + Segment 2) / 2
Altitude = (Leg 1 + Leg 2) / 2
Altitude² = Leg 1 à - Leg 2
Altitude² = (Segment 1) à - (Segment 2)
The geometric mean theorem for right triangles states that the square of the altitude drawn to the hypotenuse is equal to the product of the lengths of the two segments formed on the hypotenuse. This relation is essential in solving for unknown segments or the altitude.
In a right triangle, if one leg has length 6 and its projection on the hypotenuse measures 4, what is the length of the hypotenuse?
12
8
10
9
The property of right triangles states that the square of a leg equals the product of the hypotenuse and the projection of that leg onto the hypotenuse. Here, 6² is 36, and dividing 36 by 4 yields 9, which is the length of the hypotenuse.
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Study Outcomes

  1. Understand and explain the criteria for right triangle similarity.
  2. Apply the concept of similar triangles to solve for unknown side lengths.
  3. Analyze geometric figures to identify similar right triangles.
  4. Calculate proportional relationships and ratios in similar right triangles.
  5. Evaluate and justify the use of similarity rules in problem-solving situations.

Quick Check: Similarity in Right Triangles Cheat Sheet

  1. Define Similar Triangles - Similar triangles have the same shape but can be different sizes; their corresponding angles match and their sides stay in proportion. Think of them like two maps of the same city at different scales. Common Core: Similarity & Trigonometry
  2. Master the AA Criterion - If two angles of one triangle are congruent to two angles of another, the triangles are guaranteed to be similar because the third angles will also line up. It's a quick way to spot similarity without digging into side lengths. CCSS: Establishing AA
  3. Apply SAS Similarity - When one angle is congruent and the sides around that angle are proportional between two triangles, similarity follows. This rule helps you tackle problems where only one angle - side combo is known. Polygon Similarity Guide
  4. Use SSS Similarity - If all three corresponding sides of two triangles are proportional, the triangles must be similar, even without any angle info. It's your go‑to when you've got full side data. Polygon Similarity Guide
  5. Right Triangle Altitude Insight - Dropping the altitude to a right triangle's hypotenuse splits it into two smaller triangles, all three being similar. This neat trick unlocks a web of proportions. Effortless Math: Altitude & Similarity
  6. Observe the Geometric Mean - In a right triangle, the altitude to the hypotenuse is the geometric mean of the two segments it creates. This link powers many length and distance calculations. Effortless Math: Geometric Mean
  7. Connect Leg and Hypotenuse Mean - Each leg of a right triangle is the geometric mean between the hypotenuse and the adjacent hypotenuse segment. It's a symmetry that saves you calculation headaches. Effortless Math: Leg Means
  8. Leverage Pythagoras - Use a² + b² = c² within similar right triangles to find missing sides or to verify similarity. This classic formula shows up everywhere - embrace it! CliffsNotes: Pythagorean & Similarity
  9. Shadow Method for Heights - Compare an object's shadow to a reference stick of known height and use similar triangles to estimate building or tree heights. It's geometry meets detective work! Effortless Math: Shadow Proportions
  10. Practice Proportions Daily - Keep spotting similar right triangles and setting up side ratios to solve for unknowns. Regular practice makes these strategies second nature and quiz‑ready. Varsity Tutors: Right Triangle Similarity
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