The longer leg of a right triangle is 8 more than the shorter leg. The hypotenuse is 8 more than the longer leg. Find all three sides.
Let the shorter leg = S
The longer leg = S + 8
The Hypotenuse = S + 16
Pythagoras’s Theorem:
[S + 16]^2 = [S + 8]^2 + S^2
Solve for S:
(S + 16)^2 = S^2 + (S + 8)^2
Expand out terms of the right hand side:
(S + 16)^2 = 2 S^2 + 16 S + 64
Subtract 2 S^2 + 16 S + 64 from both sides:
-64 - 16 S - 2 S^2 + (S + 16)^2 = 0
Expand out terms of the left hand side:
-S^2 + 16 S + 192 = 0
The left hand side factors into a product with three terms:
-(S - 24) (S + 8) = 0
Multiply both sides by -1:
(S - 24) (S + 8) = 0
Split into two equations:
S - 24 = 0 or S + 8 = 0
Add 24 to both sides:
S = 24 or S + 8 = 0
Subtract 8 from both sides:
S = 24 - The short side; 24+8=32 - The longer side;
32 + 8 =40 - The Hypotenuse
Check: 40^2 = 32^2 + 24^2
1600 = 1600
