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Quizzes > High School Quizzes > Mathematics

Master Special Right Triangles Practice Quiz

Test your special right triangles knowledge and skills

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art promoting Special Triangle Showdown, a geometry trivia quiz for high school students.

In a 45-45-90 triangle, if one leg is 5, what is the hypotenuse?
5√3
7.5
5√2
10
In a 45-45-90 triangle, the hypotenuse is equal to the leg multiplied by √2. Since the leg is 5, the hypotenuse is 5√2.
In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2. If the shortest side is 4, what is the length of the hypotenuse?
4√3
6
8
4√2
For a 30-60-90 triangle, the hypotenuse is twice the length of the shortest side, so if the shortest side is 4, the hypotenuse is 8.
What are the measures of the acute angles in a 45-45-90 triangle?
60° and 60°
45° and 45°
45° and 90°
30° and 60°
In a 45-45-90 triangle, the two acute angles are both 45° as the triangle is isosceles with a right angle.
In a 30-60-90 triangle, if the hypotenuse is 10, what is the measure of the shortest side?
5√3
5
10
√10
The hypotenuse of a 30-60-90 triangle is twice the shortest side, so when the hypotenuse is 10, the shortest side is 5.
If the hypotenuse of a 45-45-90 triangle is 14√2, what is the length of each leg?
14
28
7
14√2
Each leg of a 45-45-90 triangle is found by dividing the hypotenuse by √2, so each leg is 14.
In a 30-60-90 triangle, if the shortest side is 6, what is the length of the longer leg?
6√3
3√3
2√3
12√3
The longer leg in a 30-60-90 triangle is the shortest side multiplied by √3, so if the shortest side is 6, the longer leg is 6√3.
What is the area of a 45-45-90 triangle with a leg length of 8?
32
64
16
8√2
The area of a right triangle is 1/2 times the product of its legs. For a 45-45-90 triangle with legs of 8, the area is 1/2 * 8 * 8 = 32.
A 45-45-90 triangle has an area of 50. What is the length of each leg?
√50
25
10
5√2
Since the area of a 45-45-90 triangle is 1/2 times the square of one leg, setting 1/2 * leg² = 50 gives leg² = 100, so the leg is 10.
In a 30-60-90 triangle, if the longer leg measures 12, what is the hypotenuse?
16
8√3
24
12√3
In a 30-60-90 triangle, the longer leg equals the shortest side multiplied by √3. Once the shortest side is found (12/√3), the hypotenuse, which is twice the shortest side, is 8√3.
What is the hypotenuse of a 45-45-90 triangle with legs of length 3√2?
6
3√2
6√2
9
The hypotenuse in a 45-45-90 triangle is found by multiplying a leg by √2. So, multiplying 3√2 by √2 gives 6.
If the hypotenuse of a 30-60-90 triangle is 18, what is the length of its longer leg?
6√3
9
9√3
18√3
The hypotenuse in a 30-60-90 triangle is twice the shortest side. Here, the shortest side is 9, so the longer leg, which is the shortest side multiplied by √3, is 9√3.
In a 45-45-90 triangle with an area of 72, what is the length of the hypotenuse?
24
√72
12√2
18√2
For a 45-45-90 triangle, the area is 1/2 times the square of a leg. Solving for the leg and then multiplying by √2 gives the hypotenuse as 12√2.
A 45-45-90 triangle has a hypotenuse of 10√2. What is the length of each leg?
5√2
10√2
20
10
Each leg of a 45-45-90 triangle is the hypotenuse divided by √2. Dividing 10√2 by √2 yields 10.
In a 30-60-90 triangle, which side is opposite the 60° angle?
The hypotenuse
The shortest leg
The longer leg
None of the above
In a 30-60-90 triangle, the side opposite the 60° angle is the longer leg, given the ratio 1:√3:2.
A right triangle has one angle of 45°. If one leg measures 7, what is the length of the hypotenuse?
7√2
9.9
7√3
14
In a 45-45-90 triangle, the hypotenuse is the leg multiplied by √2. Multiplying 7 by √2 gives 7√2.
A 30-60-90 triangle has an area of 27√3. Find the length of its longer leg.
3√2
6√3
9√3
9√2
Using the area formula (1/2)*x*(x√3) = 27√3 for a 30-60-90 triangle, solve for the shortest side and then multiply by √3 to get the longer leg, which is 9√2.
A 45-45-90 triangle has a perimeter of 30. What is the length of each leg in simplest radical form?
15(2 - √2)
30/(2 + √2)
10(√2 - 1)
15(2 + √2)
Let each leg be L. Then the perimeter is L + L + L√2 = L(2+√2)=30. Solving for L and rationalizing the denominator gives L = 15(2 - √2).
An isosceles right triangle and a 30-60-90 triangle have the same shorter leg length. If the area of the 30-60-90 triangle is 24√3, what is the hypotenuse of the isosceles right triangle?
8√3
4√2
4√6
4√3
For the 30-60-90 triangle, using the area formula gives the shorter leg as 4√3. Then, for the isosceles right triangle, the hypotenuse is the leg times √2, resulting in 4√3 * √2 = 4√6.
In a 30-60-90 triangle, if the difference between the hypotenuse and the longer leg is 4, what is the length of the shorter leg?
4(2 + √3)
4(2 - √3)
2 + 4√3
4√3
Let the shorter leg be x. In a 30-60-90 triangle, the hypotenuse is 2x and the longer leg is x√3. Setting up the equation 2x - x√3 = 4 and solving for x gives x = 4/(2-√3), which simplifies to 4(2+√3) after rationalization.
In a 45-45-90 triangle, if the difference between the hypotenuse and one leg is d, what is the length of the leg expressed in terms of d?
d(√2 + 1)
d(√2 - 1)
d/(√2 + 1)
d(2 - √2)
Let the leg be L. Then the hypotenuse is L√2 and the difference is L(√2 - 1)= d. Solving for L gives L = d/(√2 - 1), which, when rationalized, becomes d(√2 + 1).
0
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Study Outcomes

  1. Apply the properties of special right triangles to calculate missing side lengths and angles.
  2. Analyze geometric problems using the ratios inherent in 45-45-90 and 30-60-90 triangles.
  3. Verify the correctness of solutions by cross-checking with known special triangle relationships.
  4. Synthesize geometric concepts to approach and solve exam-style problems involving special triangles.

Special Right Triangles Cheat Sheet

  1. Properties of a 45°-45°-90° Triangle - Dive into the world of isosceles right triangles where both legs are mirror images and the hypotenuse stretches out to √2 times a leg's length. Mastering this ratio makes solving homework problems feel like a breeze. Learn more
    2. Socratic.org: Special Right Triangles
  2. Side Ratios in a 30°-60°-90° Triangle - In these golden triangles, the shortest leg is exactly half the hypotenuse, and the longer leg beams out to √3 times the shorter one. Getting comfortable with the 1:√3:2 pattern will save you tons of time on exams. Learn more
    3. Cuemath.com: Special Right Triangles Formulas
  3. Quick Ratio Recall - Memorize that 45°-45°-90° triangles hug a 1:1:√2 scheme and 30°-60°-90° triangles flaunt a 1:√3:2 pattern. Internalizing these ratios lets you whip out answers in record time and focus on trickier concepts. Quick ratios
    4. Cuemath.com: Formulas Recap
  4. Practice with Real Numbers - Try finding missing sides, such as when each leg is 5 in a 45°-45°-90° triangle and the hypotenuse magically appears as 5√2. Repetition builds confidence and cements your understanding. Practice problems
    5. OnlineMath4All: Special Right Triangles
  5. Verify Using the Pythagorean Theorem - Always double‑check your special triangle answers with a quick a² + b² = c² calculation. This extra step ensures accuracy and catches sneaky mistakes before they cost you points. Verify your work
    6. Dummies.com: Special Right Triangles
  6. Interactive GeoGebra Explorations - Jump into hands‑on GeoGebra activities to tweak angles and side lengths yourself! Seeing triangles morph in real time makes theory stick like glue. Interactive tools
    7. GeoGebra: Special Right Triangle Explorer
  7. Connect to the Unit Circle - Discover how special triangles map onto the unit circle, giving you a head start on trigonometric functions and angle measures. This bridge between geometry and trig unlocks deeper insights! Unit circle link
    8. Socratic.org: Special Right Triangles
  8. Mnemonic Devices for Easy Recall - Cook up catchy mnemonics like "1-1-√2" for isosceles rights and "1-√3-2" for 30°-60°-90° to keep those ratios top of mind. These playful memory hacks turn studying into a fun puzzle! Mnemonic tips
    9. Cuemath.com: Memory Tricks
  9. Watch Step-by-Step Video Tutorials - Sometimes you need to see it in action! Video walkthroughs break down each step so you can pause, rewind, and really grasp how those triangle sides come together. Watch videos
    10. Pearson: Special Right Triangles
  10. Test Your Skills with Quizzes - Put your new knowledge to the test with targeted quizzes that simulate exam conditions and build your confidence. Celebrate each high score as proof you're mastering special triangles! Take a quiz
    11. MathBitsNotebook.com: Practice Quizzes
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