Question

The perimeter of a right triangle is 24 centimeters. Three times the length of the longer leg minus two times the length of the shorter leg exceeds the hypotenuse by 2 centimeters. What are the lengths of all three sides?

• A. 3, 4, 5
• B. 8, 6, 10
• C. 12, 13, 5
• D. None of these

Answer 1

The correct option is B

8, 6, 10

Let the shorter leg be 2x.
The longer leg be $x^2-1$
And the hypotenuse be $x^2+1$
According to given condition.
$x^2+1+x^2-1+2x=24\Rightarrow 2x^2+2x=24\Rightarrow x^2+x-12=0....\left(i\right)$
Also,
$3(x^2-1)-2\left(2x\right)=x^2+1+2\Rightarrow 3x^2-3-4x=x^2+3\Rightarrow 2x^2-4x-6=0\Rightarrow x^2-2x-3=0.....\left(ii\right)$
Subtracting (ii) from (i), we have
$x^2+x-12-x^2+2x+3=0\Rightarrow 3x=9\Rightarrow x=3$
$\therefore$ the shorter leg = $2\times 3=6cm$
The longer leg $=(3{)}^2-1=8cm$
And the hypotenuse =$(3{)}^2+1=10cm$