Question

The longer leg of a right triangle measures 2 feet more thantwice the length of the shorter leg. The hypotenuse measures3 feet more than twice the length of the shorter leg. Find thelength of each of the three sides of the right triangle.

Answer 1

Given:

Let the length of the shorter leg be represented by x.

Hence, using Pythagoras theorem,

`\begin{gathered} \text{hypotenuse}^2=\text{adjacent}^2+\text{opposite}^2 \\ where; \\ \text{hypotenuse}=3+2x \\ \text{adjacent}=x \\ \text{opposite}=2+2x_{} \end{gathered}`

Hence,

`\begin{gathered} \text{hypotenuse}^2=\text{adjacent}^2+\text{opposite}^2 \\ (3+2x)^2=x^2+(2+2x)^2 \\ \text{Expanding the bracket,} \\ 9+12x+4x^2=x^2+4+8x+4x^2 \\ 9+12x+4x^2=5x^2+8x+4 \\ \text{Collecting the like terms to one side of the equation to form a quadratic equation,} \\ 0=5x^2-4x^2+8x-12x+4-9 \\ 0=x^2-4x-5 \\ x^2-4x-5=0 \end{gathered}`

Factorizing the quadratic equation,

Hence, the length of the three sides are;

`\begin{gathered} x=5\text{feet} \\ 2+2x=2+2(5)=2+10=12\text{feet} \\ 3+2x=3+2(5)=3+10=13\text{feet} \end{gathered}`

Therefore, the length of each of the three sides of the right triangle are;

5feet, 12 feet, and 13 feet.