Question

The perimeter of a right triangle is 24 centimeters.

Three times the length of the longer leg minus two times the length of the shorter leg exceeds the hypotenuse by 2 centimeters. What are the lengths of all three sides?​

Answer 1

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We draw a triangle, with x = shorter leg, y = longer leg, z = hypotenuse

The perimeter of a triangle is 24 centimeters.

x + y + z = 24

Three times the length of the longer leg minus two times the shorter leg exceeds the hypothenuse by 2 centimeters.

3y - 2x = z + 2

Since it's a right triangle, the Pythagorean theorem holds:

x² + y² = z²

So the system of equation is

(1) x + y + z = 24

(2) 3y - 2x = z + 2

(3) x² + y² = z²

Solve (1) for z:

(4) z = 24 - x - y

Substitute 24 - x - y for z in (2)

3y - 2x = 24 - x - y + 2

Simplify and solve for x

4y - x = 26

(5) 4y - 26 = x

Use (4) to substitute 24 - x - y for z in (3)

x² + y² = (24 - x - y)²

Square out the right side:

x² + y² = 576 + x² + y² - 48x - 48y + 2xy

Simplify

48x + 48y - 2xy = 576

Divide through by 2

(6) 24x + 24y - xy = 288

Use (5) to substitute 4y - 26 for x in (6)

24(4y - 26) + 24y - (4y - 26)y = 288

96y - 624 + 24y - 4y² + 26y = 288

-4y² + 146y - 912 = 0

Divide through by -2

2y² - 73y + 456 = 0

That factors as

(2y - 57)(y - 8) = 0

2y - 57 = 0; y - 8 = 0

2y = 57 y = 8

y = 28.5

Substitute each in (5)

4(28.5) - 26 = x; 4(8) - 26 = x

88 = x; 6 = x

We assumed that y is greater than x, since y was the longer leg.

So we can ignore y=28.5 and x=88, since that would make y the

shorter leg.

So x = 6, y = 8

Substitute in (4)

z = 24 - 6 - 8

z = 10

What are the lengths of all three legs?

They are shorter leg = x = 6, longer leg = y = 8, hypotenuse = z = 10

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