Special Right Triangles quiz

More questions about Special Right Triangles

These visible examples show the question style. LearnClash turns the full topic into ranked duels and spaced repetition.

1. 1 Easy

   In a $45^\circ-45^\circ-90^\circ$ triangle, the hypotenuse is a leg times what?

   Answer: $\sqrt{2}$

   Why: This constant ratio means the diagonal of any square is always its side length multiplied by the square root of two.

2. 2 Medium

   What is the exact sine of the $30^\circ$ angle in a special right triangle?

   Answer: $\frac{1}{2}$

   Why: This basic ratio makes the 30-degree angle one of the standard reference points on the trigonometric unit circle.

3. 3 Hard

   What is the altitude of an equilateral triangle with side length $s$?

   Answer: $\frac{s\sqrt{3}}{2}$

   Why: The altitude forms a $30-60-90$ sub-triangle, allowing you to find the exact height without using trigonometry.

4. 4 Easy

   To what total do the two acute angles in a right triangle add?

   Answer: 90°

   Why: All interior angles in a triangle total 180°, so after subtracting the 90° right angle, the remaining two must total 90°.

5. 5 Medium

   What is the perimeter of a $45^\circ-45^\circ-90^\circ$ triangle with legs of length 1?

   Answer: $2 + \sqrt{2}$

   Why: That hypotenuse is a square's diagonal, and a Babylonian clay tablet from about 1700 BC already gave it as 1.41421, right to six digits.

6. 6 Hard

   Roughly what is the smallest angle in a 5-12-13 right triangle?

   Answer: 22.6°

   Why: Niven's theorem proves no integer-sided right triangle has an acute angle measuring a rational number of degrees, so 22.6 is only a rounding.