The Perimeter of a plot... help!

Question

The perimeter of a plot of land in the shape of a right triangle is 84 miles. If the length of one leg of the triangle exceeds the other leg by 7 miles, find the length of each boundary of the land.

the lengths of the boundaries of the land are __ miles.

Answer 1

leg 1 = x

leg 2 = x + 7

perimeter = 84 miles = leg 1 + leg 2 + hypotenuse

Use pythagorean theorem to get an expression for the hypotenuse:

hypotenuse² = (leg 1)² + (leg 2)²

hypotenuse = √((leg 1)² + (leg 2)²)

hypotenuse = √((x)² + (x + 7)²) = √((x²) + (x² + 14x + 49)) = √(2x² + 14x + 49)

perimeter = leg 1 + leg 2 + hypotenuse

84 miles = (x) + (x + 7) + √(2x² + 14x + 49) = 2x + 7 + √(2x² + 14x + 49)

-2x + 77 = √(2x² + 14x + 49)

(-2x + 77)² = 2x² + 14x + 49

4x² - 308x + 5929 = 2x² + 14x + 49

0 = 2x² - 322x + 5880 = 2(x² - 161x + 2940)

0 = x² - 161x + 2940 = (x - 21)(x - 140)

x = 21, 140

Since the sides of the triangle cannot exceed the length of the perimeter, we can reject x = 140 as an answer, so x = 21.

leg 1 = x = 21 miles

leg 2 = x + 7 = 21 + 7 = 28 miles

hypotenuse = √(2x² + 14x + 49) = √(2(21)² + 14(21) + 49) = √1225 = 35 miles

Answer 2

Hi Amber. The easiest way to solve this is to remember it’s a right triangle. So it could be a multiple of a 3-4-5 right triangle. Since the difference between the first and second leg is 7, we should multiply the 3-4-5 by 7, as it gives us a difference of 7 for each side. That would give us 21-28-35, which totals 84. So 21,28, and 35 are the three sides.

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