Question

Help please

Find the length of the shorter leg of a right triangle if the longer leg is 7 feet more than the shorter leg and the hypotenuse is 5 feet less than twice the shorter leg.


The length of the shorter leg is ____

Answer 1

17.7 ft

Explanation:

Let x be the shorter leg

Longer: x + 7

Hypotenuse: 2x - 5

(2x - 5)² = (x + 7)² + x²

4x² - 20x + 25 = x² + 14x + 49 + x²

2x² - 34x - 24 = 0

x² - 17x - 12 = 0

x = [17 +- sqrt(17² - 4(1)(-12)]/2

x = [17 + sqrt(337)]/2

x = 17.67877988

Answer 2

17 ± sqrt(337)

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2

Explanation:

Let a = shorter leg

b= longer leg = a+7

c = hypotenuse = 2a-5

We can use the Pythagorean theorem

a^2+b^2 = c^2

a^2 + (a+7)^2 = (2a-5)^2

(a+7)^2 = a^2 +7a +7a+49 = a^2 +14a +49

(2a-5)^2 = 2a*2a -10a -10a +25 = 4a^2 -20a +25

Substituting these into the equation

a^2 + (a^2 +14a +49) = (4a^2 -20a +25)

Combine like terms

2a^2 +14a +49 = 4a^2 -20a +25

Subtracting 2a^2 from each side

2a^2 -2a^2 +14a +49 = 4a^2-2a^2 -20a +25

+14a +49 = 2a^2 -20a +25

Subtract 14a from each side

-14a+14a +49 = 2a^2 -20a -14a+25

+49 = 2a^2 -34a +25

Subtract 49 from each side

49-49 = 2a^2 -34a +25-49

0 = 2a^2 -34a -24

Divide each side by 2

0/2 = 2/2a^2 -34/2a -24/2

0 = a^2 -17a -12

Using the quadratic formula

a=1 b= -17 c = -12

17 ±sqrt( (-17)^2 - 4(1)(-12))

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2(1)

17 ± sqrt(337)

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2