Question

the longer leg of a right triangle is 1 ft longer than the shorter leg. the hypotenuse is 9 ft longer than the shorter leg. find the side lengths of the triangle. length of the shorter leg:length of the longer leg:length of the hypotenuse:

Answer 1

length of the shorter leg: 20 ft

length of the longer leg: 21 ft

length of the hypotenuse: ​29 ft

The hypotenuse refers to the longest side of a triangle

Let the shorter leg of the triangle in this case be x ft

Since the longer leg is 1 ft longer than the shorter leg, then the longer leg will be (x+1) ft

Furthermore, we are told that the hypotenuse of the triangle is 9 ft longer than the shorter leg, what this means is that the length of the hypotenuse is (x+9) ft

From Pythagoras' theorem, the square of the length of the hypotenuse is equal the sum of the squares of the two other sides

With respect to the question at hand;

`\begin{gathered} (x+9)^2=x^2+(x+1)^2 \\ \\ \text{Expanding the brackets, we have} \\ \\ x^2+18x+81=x^2+x^2\text{ + 2x + 1} \\ \\ x^2+18x+81=2x^2\text{ + 2x + 1} \\ \\ 2x^2-x^2\text{ + 2x-18x + 1 -81 = 0} \\ \\ x^2\text{ - 16x - 80 = 0} \\ \\ x^2\text{ - 20x + 4x - 80 = 0} \\ \\ x(x-20)\text{ + 4(x-20) = 0} \\ \\ (x+4)(x-20)\text{ = 0} \\ x\text{ + 4 = 0 or x - 20 = 0} \\ \\ x\text{ = -4 or 20} \end{gathered}`

Since the length of a triangle cannot be negative, then the length in this case is 20 ft

The length of the shorter side x = 20 ft

The length of the longer side = x + 1 = 20 + 1 = 21 ft

The length of the hypotenuse = x + 9 = 20 + 9 = 29 ft