# 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 8 , 7 , and 14 , respectively. What is the rectangle's area?

Oct 30, 2016

$54.18$

#### Explanation:

Draw the diagram

To find the are of the triangle at the bottom,

We need to find the length of $\text{B}$

As the triangle is a right angle triangle,

We use the color(blue)("Pythagoras theorem"

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

Where,

color(orange)("c = Longest side (hypotenuse)"

color(orange)("a and b are the other sides"

So,

color(purple)(c=8

color(purple)(b="B"

color(purple)(a=7

Insert the following (above) in the formula

$\rightarrow {7}^{2} + {\text{B}}^{2} = {8}^{2}$

$\rightarrow 49 + {\text{B}}^{2} = 64$

$\rightarrow {\text{B}}^{2} = 64 - 49$

$\rightarrow {\text{B}}^{2} = 15$

Take the square root of both sides

$\rightarrow \sqrt{{\text{B}}^{2}} = \sqrt{15}$

color(green)(rArr"B"=sqrt15~~3.87

Let's redraw the diagram

Now, we need to find the area of the rectangle

We use the formula

color(brown)("Area of rectangle"=l*b

Where,

color(orange)(l="length"=14

color(orange)(b="breadth"=sqrt15

$\rightarrow l \cdot b$

$\rightarrow 14 \cdot \sqrt{15}$

$\rightarrow 14 \sqrt{15}$

$\rightarrow 14 \cdot 3.87$

color(green)(rArr54.18