Question

A right triangle has sides A, B, and C. Side A is the hypotenuse and side B is also a side of a rectangle. Sides A, C, and the side of the rectangle adjacent to side B have lengths of `13`, `4`, and `3`, respectively. What is the rectangle's area?

Answer 1

`9sqrt17`

Explanation:

Consider the diagram

We can find the length of `b` using the Pythagoras-theorem

`color(blue)(a^2+b^2=c^2`

Where,

`aand b` are the right-containing sides and `c` is the Hypotenuse (longest side of a right-triangle)

But,

In this case,`a` is the Hypotenuse and `band c` are the right-containing sides

So,

`rarrb^2+4^2=13^2`

`rarrb^2+16=169`

`rarrb^2=169-16`

`rarrb^2=153`

`rarrb=sqrt153`

`color(green)(rArrb=sqrt153=sqrt(9*17)=3sqrt17`

Now we need to find the area of the rectangle

Area=`color(purple)(l*b`

Where,

`l=l` `eng` `th,b=breadth`

`color(violet)(l=3sqrt17,b=3`

`rarrArea=3sqrt17*3`

`color(green)(rArr9sqrt17`