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Quizzes > Mathematics

Special Right Triangles Quiz for SAT Geometry

Quick, free 30-60-90 practice. Instant results.

Editorial: Review CompletedCreated By: Spl ProductionUpdated Aug 24, 2025
Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Paper art illustrating a trivia quiz for high school geometry, focusing on right triangles.

This special right triangles quiz helps you practice 30-60-90 and 45-45-90 skills for SAT geometry, check trig ratios, and spot gaps in 20 quick questions. For extra practice, check the right triangle trigonometry quiz, review legs and hypotenuse with the pythagorean theorem test, and connect ideas in the right triangle similarity quiz.

What is the defining property of a right triangle?
It has two 90° angles
All angles are acute
It has one 90° angle
It has one angle equal to 60°
A right triangle always contains exactly one 90° angle, which distinguishes it from other types of triangles. This 90° angle is the key defining feature.
In a 45-45-90 triangle, if one leg is 5 units long, what is the length of the hypotenuse?
5√2
5√3
5
10
In a 45-45-90 triangle, the hypotenuse is equal to the leg multiplied by √2. Therefore, if one leg is 5 units, the hypotenuse is 5√2.
What is the side ratio of a 30-60-90 triangle?
1 : 2 : √3
√3 : 1 : 2
1 : √3 : 2
2 : √3 : 1
The sides of a 30-60-90 triangle have the ratio 1 : √3 : 2, corresponding to the sides opposite the 30°, 60°, and 90° angles respectively. This ratio is a standard property of such triangles.
Which theorem is used to determine the relationship between the sides of a right triangle?
Congruence Postulate
Pythagorean theorem
Alternate Interior Angles Theorem
Triangle Sum Theorem
The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is fundamental for working with right triangles.
If a right triangle has legs of 3 and 4 units, what is the length of the hypotenuse?
7
6
8
5
Using the Pythagorean theorem, 3² + 4² equals 9 + 16 which is 25. The square root of 25 is 5, making the hypotenuse 5 units long.
In a 45-45-90 triangle with a hypotenuse of 10 units, what is the length of one leg?
5√2
10
5
√10
For a 45-45-90 triangle, each leg is found by dividing the hypotenuse by √2. Hence, with a hypotenuse of 10, one leg is 10/√2, which simplifies to 5√2.
What are the measures of the acute angles in an isosceles right triangle?
30° and 60°
60° and 60°
45° and 90°
45° and 45°
An isosceles right triangle has one 90° angle, with the remaining two angles being equal. Since these two angles must add up to 90°, they each measure 45°.
In a 30-60-90 triangle, if the longer leg measures 8√3 units, what is the length of the shorter leg?
8√3
16
8
4
In a 30-60-90 triangle, the longer leg is √3 times the length of the shorter leg. Dividing the longer leg of 8√3 by √3 gives the shorter leg as 8.
A right triangle has one leg of length 6 and a hypotenuse of length 10. What is the length of the other leg?
6
4
10
8
By applying the Pythagorean theorem (6² + b² = 10²), we find b² = 100 - 36 = 64, which gives b = 8.
If the sides of a right triangle are in arithmetic progression, which triangle is it?
3-4-5 triangle
30-60-90 triangle
2-3-4 triangle
45-45-90 triangle
The 3-4-5 triangle is the classic example of a right triangle where the side lengths form an arithmetic progression with a common difference. The other sets do not meet this criterion.
Which of the following sets of side lengths can represent a right triangle?
7, 8, 9
2, 3, 4
4, 5, 6
5, 12, 13
Only the set 5, 12, 13 satisfies the Pythagorean theorem (5² + 12² = 13²), making it a valid set of side lengths for a right triangle.
In a right triangle, if one of the acute angles measures 30°, which side is opposite the 30° angle?
The shortest side
The side adjacent to the 90° angle
The hypotenuse
The longest side
In a 30-60-90 triangle, the side opposite the 30° angle is conventionally the shortest side. This is a defining characteristic of the triangle.
In a right triangle with legs of lengths 7 and 24, what is the length of the hypotenuse?
31
24
25
7
Using the Pythagorean theorem, 7² + 24² equals 49 + 576 = 625, and the square root of 625 is 25. Thus, the hypotenuse is 25 units.
If a right triangle's acute angles are in the ratio 1:2, what are the measures of these angles?
15° and 30°
45° and 45°
20° and 40°
30° and 60°
Since the two acute angles in a right triangle must add up to 90° and they are in the ratio 1:2, the angles must be 30° and 60°.
What is the length of the hypotenuse in a 30-60-90 triangle with a short leg measuring 9 units?
9
18
9√3
9√2
In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg. Therefore, if the short leg is 9 units, the hypotenuse is 18 units.
A 45-45-90 triangle and a 30-60-90 triangle have the same hypotenuse length, H. If the leg of the 45-45-90 triangle is L and the shorter leg of the 30-60-90 triangle is S, what is the ratio L:S?
√3 : 1
√2 : 1
1 : √2
1 : 2
For the 45-45-90 triangle, L = H/√2, and for the 30-60-90 triangle, S = H/2. Dividing these gives (H/√2) ÷ (H/2) = 2/√2, which simplifies to √2:1.
Determine the area of a right triangle that is known to be a 45-45-90 triangle with one leg measuring 5√2 and a hypotenuse of 10 units.
10√2
50
12.5
25
In a 45-45-90 triangle both legs are equal. The area is half the product of the legs: ½ Ã- (5√2) Ã- (5√2) = ½ Ã- 50 = 25.
The altitude drawn to the hypotenuse of a right triangle divides it into two smaller right triangles that are similar to the original triangle. Which theorem describes this property?
Isosceles Triangle Theorem
Pythagorean Theorem
Altitude-on-Hypotenuse Theorem
Parallel Postulate
The Altitude-on-Hypotenuse Theorem, also known as the Geometric Mean Theorem, explains that the altitude to the hypotenuse divides the triangle into two smaller triangles that are similar to the original.
Solve for x if the sides of a right triangle are expressed as: leg = x, other leg = x+2, and hypotenuse = x+4.
4
-2
6
8
Using the Pythagorean theorem, we set up the equation x² + (x+2)² = (x+4)² which simplifies to x² - 4x - 12 = 0. Solving this quadratic gives x = 6 (rejecting the negative solution).
A right triangle has legs k and k√3, and a hypotenuse of 2k. If the area of this triangle is 16√3, find the value of k.
4√2
8
2√2
4√3
The area of the triangle is given by ½ Ã- k Ã- (k√3) = (k²√3)/2. Setting this equal to 16√3 gives k²/2 = 16, so k² = 32 and thus k = 4√2.
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Study Outcomes

  1. Analyze the properties of right triangles.
  2. Apply the Pythagorean theorem to calculate unknown side lengths.
  3. Solve problems involving trigonometric ratios in right triangles.
  4. Identify and use the defining characteristics of special right triangles.
  5. Synthesize geometric concepts to solve right triangle problems.

Right Triangle Trigonometry Cheat Sheet

  1. Master the 45°-45°-90° Triangle - In this special right triangle both legs are the same length and the hypotenuse is √2 times a leg, making calculations a snap. Imagine slicing a square corner-to-corner to see this perfect half-square shape.
  2. Recognize the 30°-60°-90° Ratios - The shortest side (opposite 30°) is x, the longer leg (opposite 60°) is x√3, and the hypotenuse is 2x. Picture an equilateral triangle chopped in half - it reveals these neat proportions every time.
  3. Memorize SOH‑CAH‑TOA - Use this catchy mnemonic to recall Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent without a second thought. It's like having a secret code that unlocks any right‑triangle problem.
  4. Apply the Pythagorean Theorem - Verify side lengths with a² + b² = c² and turn guesswork into solid proof. It's the ultimate check for right triangles and a great way to ensure your numbers add up.
  5. Practice Missing Side Problems - Grab random leg or hypotenuse values and use your special-triangle ratios to solve for the unknown side. With steady practice, you'll spot patterns faster than you can say "√3."
  6. Spot the Isosceles Right Triangle - Know that the 45°‑45°‑90° triangle is just an isosceles right triangle with two equal angles and legs. This insight helps you connect geometry concepts and ace those proofs.
  7. Derive from an Equilateral Triangle - See how dropping an altitude in an equilateral triangle creates two 30°-60°-90° triangles, revealing the golden side ratios. This visual trick cements your understanding in one elegant step.
  8. Simplify Trig Without a Calculator - Use the known ratios of 45°-45°-90° and 30°-60°-90° to find sine, cosine, and tangent values instantly. It's like having a cheat sheet built right into your brain!
  9. Remember Leg Positions in 30°-60°-90° - Always place the longer leg opposite the 60° angle and the shorter leg opposite 30°. Keeping this straight helps you solve problems in record time.
  10. Apply to Real-World Problems - Use these triangles to calculate heights, distances, and even the pitch of a roof. Turning textbook theory into practical skills makes math stick and keeps it fun!
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