Respuesta :

Answer:

45.7 square feet

Step-by-step explanation:

Given: Dimensions of triangle are 16 feet, 12 feet and 20 feet

To find: exact area in square feet of the flower bed NOT including the fountain

Solution:

A triangle in which square of hypotenuse is equal to sum of squares of the base and perpendicular is a right angled triangle.

Here, [tex](hypotenuse)^2 =20^2=400[/tex]

[tex](base)^2+(perpendicular)^2=16^2+12^2=256+144=400[/tex]

As [tex](base)^2+(perpendicular)^2=(hypotenuse)^2[/tex], the given triangle is a right angled triangle.

So, radius of the incircle is given by [tex]\frac{(perpendicular+base-hypotenuse)}{2}=\frac{16+12-20}{2} =\frac{8}{2}=4[/tex] feet

Area of circle [tex]=\pi(radius)^2\\[/tex]

[tex]=\frac{22}{7}(4)^2\\= 50.3[/tex] square feet

Area of triangle=[tex]\frac{1}{2}(base)(height)[/tex]

Put base = 12 feet and height = 16 feet

area of triangle = [tex]\frac{1}{2}(12)(16)=96[/tex] square feet

So, area in square feet of the flower bed NOT including the fountain = 96-50.3=45.7 square feet

We have been given a flower bed that is in form of right triangle. We are asked to find the area of flower bed without the fountain.

First of all, we need to find the radius of circle.  

When a circle is inscribed inside a right triangle with sides a, b and c (hypotenuse), then the radius can be found by formula:

[tex]r=\frac{a+b-c}{2}[/tex]

Upon substituting our given values in above formula, we will get:

[tex]r=\frac{16+12-20}{2}[/tex]

[tex]r=\frac{8}{2}=4[/tex]

The area of the flower bed will be area of right triangle minus area of fountain.

[tex]\text{Area of flower bed without fountain}=\text{Area of right triangle}-\text{Area of circle}[/tex]

[tex]\text{Area of flower bed without fountain}=\frac{1}{2}\cdot 12\text{ ft}\cdot 16\text{ ft}-\pi (4\text{ ft})^2[/tex]

[tex]\text{Area of flower bed without fountain}=6\cdot 16\text{ ft}^2-16\pi \text{ ft}^2[/tex]

[tex]\text{Area of flower bed without fountain}=96\text{ ft}^2-16\pi \text{ ft}^2[/tex]

Therefore, the exact area of the flower bed without fountain is [tex]96-16\pi[/tex] square feet.