A rectangle is drawn so the width is 8 inches longer than the height. If the rectangle's diagonal measurement is 57 inches, find the height.

Answer 1

We need the Pythagorean Theorem to solve for the height of the rectangle.

a² + b² = c²

The rectangle's diagonal represents the hypotenuse of the right triangle: c = 57

The width of the rectangle represents one of the legs of the right triangle: a = x + 8

The height of the rectangle represents one of the legs of the right triangle: b = x

(x + 8)² + x² = 57²

x² + 8x + 8x + 64 + x² = 3249

2x² + 16x + 64 = 3249

2x² + 16x - 3185 = 0

Use the quadratic formula to solve for x. Let a = 2, b = 16, and c = -3185.

x = [-b ± √(b² - 4ac)] / 2a

x = [-16 ± √(16² - 4(2)(-3185))] / 2(2) ⇒ x = [-16 ± √(256 + 25,480)] / 4 ⇒ x = [-16 ± √(25,736)] / 4

⇒ x ≈ -44.11 or x ≈ 36.11 (negative solution doesn't exist when measuring)

The height of the rectangle is about 36.11 inches.