A rectangular painting has a diagonal measure of 17 inches and an area of 120 square inches. Use the formula for the area of a rectangle and the Pythagorean Theorem to find the length and width of the painting.

Length = 8 inches

Width = 15 inches

Step-by-step explanation:

A rectangular painting has a diagonal measure of 17 inches and an area of 120 square inches. Use the formula for the area of a rectangle and the Pythagorean Theorem to find the length and width of the painting.

Let:

a = length of painting

b = width of painting

Area of painting = 120 = ab

Pythagorean Theorem says

a² + b² = Diagonal²

a² + b²= 17²

[120 =ab means b= 120/a]

a² + (120/a)² = 17²

Multiply through by a²

a⁴ + 14400 = 289

z² + 14400 = 289z [for now, substitute z for a²]

z² - 289z + 14400 = 0

z² - 64z - 225z + 14400 = 0

z(z - 64) -225(z - 64) = 0

(z - 64)(z - 225) = 0

z - 64 = 0

z = 64

z - 225 = 0

z = 225

Note that:

z for a²

Hence:

a² = 64

a = √64

a = 8 inches

The same applies for b

b² = 225

b = √225

b = 15 inches

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