Question

The hypotenuse of a right triangle is 5 m long. The shorter leg is 1 m shorter than the longer leg. Find the side lengths of the triangle.mLength of the shorter leg: -Length of the longer leg: 0Length of the hypotenuse: 0m

Answer 1

To answer this question we will use the Pythagorean theorem.

Let l be the length (in meters) of the longer leg, then the length of the shorter (in meters) leg will be l-1.

Using the Pythagorean theorem we get:

`l^2+(l-1)^2=5^2.`

Simplifying the above result we get:

`l^2+l^2-2l+1=25.`

Adding like terms we get:

`2l^2-2l+1=25.`

Subtracting 25 from the above equation we get:

`\begin{gathered} 2l^2-2l+1-25=25-25, \\ 2l^2-2l-24=0. \end{gathered}`

Dividing the above equation by 2 we get:

`\begin{gathered} (2l^2-2l-24)/(2)=(0)/(2), \\ l^2-l-12=0. \end{gathered}`

Now, notice that:

`\begin{gathered} -1=3-4, \\ -12=3*(-4). \end{gathered}`

Then:

`l^2-l-12=(l+3)(l-4).`

Therefore:

`l^2-l-12=0\text{ if and only if }l=4\text{ or l=-3.}`

Since l is the length of a leg of a right triangle, then l>0, therefore l=4.

Answer:

Length of the shorter leg: 4m.

Length of the longer leg: 3m.

Length of the hypotenuse: 5m.