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?

Apr 11, 2016

$9 \sqrt{17}$

Explanation:

Consider the diagram

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

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

Where,

$a \mathmr{and} b$ 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 $b \mathmr{and} c$ are the right-containing sides

So,

$\rightarrow {b}^{2} + {4}^{2} = {13}^{2}$

$\rightarrow {b}^{2} + 16 = 169$

$\rightarrow {b}^{2} = 169 - 16$

$\rightarrow {b}^{2} = 153$

$\rightarrow b = \sqrt{153}$

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$$e n g$$t h , b = b r e a \mathrm{dt} h$

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

$\rightarrow A r e a = 3 \sqrt{17} \cdot 3$

color(green)(rArr9sqrt17