The longer leg of a right triangle is 3inches longer than the shorter leg.

Question

The longer leg of a right triangle is
3inches
longer than the shorter leg. The hypotenuse is
6inches
longer than the shorter leg. Find the side lengths of the triangle.
shorter leg-
longer leg=
hypotenuse=

Answer 1

Hi Key,

 

Lets start by labeling the sides. Say the smaller leg is x

 

=> this implies that the longer leg is x + 3

 

=> and that the hypotenuse is x + 6

 

You know that the sum of the square of the two legs equals the square of the hypotenuse a^2 + b^2 = c^2

 

a = x, b = x + 3, c = x + 6.

 

=> x^2 + (x + 3)^2 = (x + 6)^2

 

<=> x^2 + x^2 + 6x + 9 = x^2 + 12x + 36

 

Now you move everything to the left

 

2x^2 - x^2 + 6x - 12x + 9 - 36 = 0 

 

<=> x^2 - 6x -27 = 0

 

 

Answer 2

We don't know the length of the shorter leg of the triangle. So let we call it X.  Or:  S= X  (S stands for shorter)

 

The longer leg is 3 inches longer than the shorter leg. So:  L= 3+X   (whatever X is, L is 3 inches longer).

 

Hypotenuse is 6 inches longer than the shorter leg. So:  H=X+6

 

Now, for a right triangle, we can apply Pythagorean Theorem:

 

H^2 = L^2 + S^2

 

so: (x+6)^2 = X^2 + (X+3)^2

 

just expand, and you get:  X^2 + 12X + 36 = X^2 + X^2 + 6X + 9

 

So: X^2 - 6X -27 = 0   => (X-9) (X+3) = 0    => X= 9  or X=-3

 

Since length can not be negative, only X+9 is acceptable. And since in the beginning we said S=X, therefore we have found the shorter side to be 9. 

 

Quick check: If S=9, L= 3+9=12 and H= 6+9= 15

 

15^2=225

12^2=144

9^2=81

 

81+144=  225 Which means we have done it correctly!