A frequent concept on number sense tests deals with properties of right triangles. This concept appeared 18 times last year in high school, with a median placement at question # 45.
Number Dojo Level: 253
Many people are familiar with the Pythagorean Theorem, which states that the square of the length of the longest side (hypotenuse) of a right triangle is equal to the sum of the squares of the lengths of the shorter two sides (legs). Put more simply:
It will be helpful to memorize a few Pythagorean triples--where the side lengths are integers and form a right triangle. A primitive Pythagorean triple is one where the side lengths are "coprime"--meaning they have no common factors other than 1. There are 16 such triples with sides < 100; here are 10 of them:
- 3, 4, 5
- 5, 12, 13
- 7, 24, 25
- 8, 15, 17
- 9, 40, 41
- 11, 60, 61
- 12, 35, 37
- 13, 84, 85
- 16, 63, 65
- 20, 21, 29
- 6, 8, 10
- 10, 24, 26
- 14, 48, 50
- 16, 30, 34, etc.
Special Right Triangles
- A 30°-60°-90° triangle has side lengths in a ratio of 1: √3: 2. In other words, if the shorter leg (opposite the 30° angle) has length 1, the longer leg (opposite the 60° angle) has length √3, and the hypotenuse (opposite the 90° angle) has length 2.
- A 45°-45°-90° triangle is also called an isosceles right triangle because its legs have equal length. The sides have a ratio of 1: 1: √2. In other words, if the legs (opposite the 45° angles) each have length 1, the hypotenuse (opposite the 90° angle) has a length √2.
Altitude to the Hypotenuse
If you draw a line from the vertex of the right angle to the hypotenuse--and form two right angles where the line meets the hypotenuse--you have drawn its altitude. In other words, if you turn the triangle such that its hypotenuse is the base, you have drawn its height.
To calculate the length of this altitude, simply use the formula ab/c, where a & b are the legs of the original triangle, and c is the original hypotenuse.
How to Solve:
- Based on the information given, determine which type of right triangle you have.
- Perform the necessary calculations to determine the unknown, which may be a side length, the perimeter, the area, or the altitude to the hypotenuse.
Example 1: If a right triangle has a hypotenuse of 41 and one leg of 40, then the other leg is ___
- Look at the list of Pythagorean Triples above. The relevant triple is 9, 40, 41.
- The answer is 9.
Example 2: The leg opposite the 30° angle in a right triangle is 6 inches. The hypotenuse is ___ inches.
- We have a 30°-60°-90° triangle. The hypotenuse will be 2x the shorter leg.
- 6 x 2 = 12.
Example 3: The area of a right triangle with base of 7 and a hypotenuse of 25 is ___
- Look at the list of Pythagorean Triples above. The relevant triple is 7, 24, 25.
- The area = bh/2, or 7(24)/2, which is 84.
Example 4: The perimeter of a right triangle with legs of 5 and 12 is ___
- Look at the list of Pythagorean Triples above. The relevant triple is 5, 12, 13.
- The perimeter is the sum of the sides. 5 + 12 + 13 = 30.
Example 5: The legs of a right triangle are 3 and 4. The length of the altitude to the hypotenuse is ___ (decimal).
- This is the simplest Pythagorean Triple: 3, 4, 5.
- The altitude to the hypotenuse is ab/c, or (3 x 4)/5, which is 12/5 or 2.4.
Example 6: The area of an isosceles right triangle with a hypotenuse length 12√2 is ___
- We have a 45°-45°-90° triangle. Each leg will be the hypotenuse divided by √2. So each leg is 12.
- The area is bh/2, or 12(12)/2, which is 72.