Question

One leg of a right triangle has a length of 15 m. the other sides have lengths that are consecutive integers. find the number of meters in the perimeter.

Answer 1

$\text{Let one leg of a right triangle has length of 15 m.}\\ \\ \text{And the other two sides have lengths that are consecutive integers.}\\ \\ \text{let the length of other leg of right triangle is x,}\\ \text{then the length of the hypotenus of the right triangle would be}=x+1\\ \\ \text{Now in the right triangle, using the Pythagorean theorem, we have}\\ \\ \text{(Hypotenus)}^2=(\text{adjacent})^2+(\text{opposite})^2$

$\Rightarrow (x+1)^2=(15)^2+(x)^2\\ \\ \Rightarrow x^2+2x+1=225+x^2\\ \\ \Rightarrow x^2+2x-x^2=225-1\\ \\ \Rightarrow 2x=224\\ \\ \Rightarrow x=\frac{224}{2}\\ \\ \Rightarrow x=112\\ \\ \text{Hence the other leg of right triangle is 112 meters,}\\ \text{and hypotenus is 113 meters.}\\ \\ \text{Therefore, the perimeter of the right triangle}=15+112+113=240 \text{ meters}$

Hence, Perimeter of the right triangle is: 240 meters