solve this please

Question

The hypotenuse of a right triangle is


5ft
long. The shorter leg is


1ft
shorter than the longer leg. Find the side lengths of the triangle.

 

Length of the shorter leg:





ft


Length of the longer leg:





ft


Length of the hypotenuse:





ft

Answer 1

First, I’d recommend learning to translate words into precise, concise math notation:

For a right triangle with sides A, B, and C:

“The hypotenuse of a right triangle is 5ft long” means C = 5ft

“The shorter leg is 1ft shorter than the longer leg” means A + 1 = B

Now, the math:
A² + B² = C² [ Pythagorean Theorem]
A² + (A+1)² = 25
A² + A² + 2A + 1 = 25
2A² + 2A – 24 = 0

So,
A = (-2 + SQRT(4+192))/4 or A = (-2 – SQRT(4+192))/4
A = 3 or A = -4 (but A must be positive)

And
A + 1 = B
4 = B

Checking:
Is 3² + 4² = 5² ? yes.

p.s., this triangle -- 3x, 4x, 5x – will appear in lots of problems and on lots of tests!

Answer 2

5² = x² + (x - 1)²

25 = x² + x² - 2x + 1

0 = 2x² - 2x - 24

0 = 2(x² - x - 12)

0 = x² - x - 12

0 = (x - 4)(x + 3)

 

{-3, 4}

 

Lengths cannot be negative. The long leg is 4 and the short let is 3.