**High School Number Sense Lesson 44: Right Triangles**

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:

**c**

^{2}= a^{2}+ b^{2}**Pythagorean Triples**

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**.

**Here's a free worksheet to help you practice TriangleRight:**

triangleright.pdf |

**Up Next for High School: SolveXDenom**