Math

QuestionThe shorter leg is 8ft8 \mathrm{ft} less than the longer leg, and the hypotenuse is 8ft8 \mathrm{ft} more than the longer leg. Find the lengths.

Studdy Solution

STEP 1

Assumptions1. The shorter leg of the triangle is 8ft8 \mathrm{ft} shorter than the longer leg. . The hypotenuse is 8ft8 \mathrm{ft} longer than the longer leg.
3. The triangle is a right triangle, so we can use the Pythagorean theorem to solve the problem.

STEP 2

Let's denote the length of the shorter leg as aa, the length of the longer leg as bb, and the length of the hypotenuse as cc.

STEP 3

According to the problem, we have the following relationshipsa=b8a = b -8c=b+8c = b +8

STEP 4

Since the triangle is a right triangle, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. We can write this asa2+b2=c2a^2 + b^2 = c^2

STEP 5

Substitute the expressions for aa and cc from steps3 into the equation from step4.
(b8)2+b2=(b+8)2(b -8)^2 + b^2 = (b +8)^2

STEP 6

Expand the squares on both sides of the equation.
b216b+64+b2=b2+16b+64b^2 -16b +64 + b^2 = b^2 +16b +64

STEP 7

implify the equation by combining like terms.
2b216b=b2+16b2b^2 -16b = b^2 +16b

STEP 8

Subtract b2b^2 and 16b16b from both sides of the equation to isolate bb.
b232b=0b^2 -32b =0

STEP 9

Factor out bb from the equation.
b(b32)=b(b -32) =

STEP 10

Set each factor equal to zero and solve for bb.
b=0,b=32b =0, b =32

STEP 11

Since the length of a side of a triangle cannot be zero, we discard b=0b =0 and take b=32b =32.

STEP 12

Substitute b=32b =32 into the equations from step to find aa and cc.
a=328=24a =32 -8 =24c=32+8=40c =32 +8 =40So, the lengths of the sides of the triangle are 24ft24 \mathrm{ft}, 32ft32 \mathrm{ft}, and 40ft40 \mathrm{ft}.

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