Question

A right triangle has two shorter sides that differ in length by 7cm. The length of the

hypotenuse is 8 cm longer than the shortest side. Find the lengths of the three sides.
Show all of your steps.

Pls help!!! 75 points

Answer 1

The lengths of the sides are 5, 12, 13.

Explanation:

In a right triangle, the two shorter sides are the legs. The longest side is the hypotenuse.

Let the shorter leg = x.

The longer leg is 7 cm longer, so its length is x + 7.

The length of the hypotenuse is 8 cm longer than the shorter leg, so its length is x + 8.

The lengths are:

x, x + 7, x + 8

Since the triangle is a right triangle, we can use the Pythagorean theorem.

a^2 + b^2 = c^2

x^2 + (x + 7)^2 = (x + 8)^2

Square the trinomials.

x^2 + x^2 + 14x + 49 = x^2 + 16x + 64

Combine like terms and place them all on the left side equaling zero.

x^2 - 2x - 15 = 0

Factor the left side.

(x - 5)(x + 3) = 0

x - 5 = 0 or x + 3 = 0

x = 5 or x = -3

Since the length of a side of a triangle cannot be negative, we discard the solution x = -3.

x = 5

x + 7 = 5 + 7 = 12

x + 8 = 5 + 8 = 13

Answer: The lengths of the sides are 5, 12, 13.

Answer 2

a = 5

b = 12

c = 13

Explanation:

a^2+b^2=c^2

b-a=7(b=a+7)

c=a+8

Then, substitute

a^2+((a+7)*(a+7))=c^2

a^2+a^2+7a+7a+49=c^2

2a^2+14a+49=c^2

Because c = a+8

2a^2+14a+49=(a+8)(a+8)

2a^2+14a+49=a^2+16a+64

a^2-2a=15

a^2-2a-15=0

(a-5)(a+3)=0

a = 5,-3

a = 5(a side cannot be negative)

Plug in a=5 to the other equations to get

a = 5, b = 12, c = 13

Hope it helps <3