Grade 8 Radicals: Pythagorean Theorem

Worked examples

Real problems from the practice worksheets, solved one step at a time.

1. Example 1

   A right triangle has a hypotenuse of 11 units and one leg of 5 units. Find the length of the other leg to the nearest tenth.

   1. In a right triangle, a² + b² = c², where c is the hypotenuse — the longest side, across from the right angle.
   2. b² = 11² − 5² = 121 − 25 = 96.
   3. b = √96.
   4. 9² = 81 and 10² = 100, so √96 is between 9 and 10.
   5. Try tenths: 9.7² = 94.09 and 9.8² = 96.04, so √96 is between 9.7 and 9.8.
   6. Halfway is 9.75, and 9.75² = 95.0625. 96 is more than 95.0625, so √96 is closer to 9.8.
   7. So the other leg is about 9.8 units (to the nearest tenth).

   Answer: 9.8 units

2. Example 2

   A rectangle is 18 in wide and 24 in long. How long is its diagonal (in inches)?

   1. The diagonal splits the rectangle into two right triangles: the width and length are the legs, and the diagonal is the hypotenuse.
   2. d² = 18² + 24² = 324 + 576 = 900.
   3. d = √900 = 30, because 30 × 30 = 900.
   4. So the diagonal is 30 inches.

   Answer: 30

3. Example 3

   A rectangle is 40 ft wide and 9 ft long. How long is its diagonal (in feet)?

   1. The diagonal splits the rectangle into two right triangles: the width and length are the legs, and the diagonal is the hypotenuse.
   2. d² = 40² + 9² = 1600 + 81 = 1681.
   3. d = √1681 = 41, because 41 × 41 = 1681.
   4. So the diagonal is 41 feet.

   Answer: 41