Answer:
[tex]3.9 in^2[/tex]
Step-by-step explanation:
The area of a triangle is given by
[tex]A=\frac{1}{2}bh[/tex]
where
b is the base
h is the height
Here we have an equilateral triangle, which has the 3 sides of the same length.
Let's call L the length of one side.
We know that the perimeter of the triangle is
p = 9 in
The perimeter is the sum of the three sides, so:
[tex]p=L+L+L=3L[/tex]
Therefore, we find the length of the side:
[tex]L=\frac{p}{3}=\frac{9}{3}=3 in[/tex]
Therefore the length is the base of the triangle,
[tex]b=L[/tex]
The height can be calculated by considering half triangle: the hypothenuse is equal to L, while one side is equal to half the base (b/2), therefore the height is given by Pythagorean's theorem:
[tex]h=\sqrt{L^2-(\frac{b}{2})^2}=\sqrt{3^2-(\frac{3}{2})^2}=2.6 in[/tex]
Therefore, the area of the triangle is:
[tex]A=\frac{1}{2}(3)(2.6)=3.9 in^2[/tex]
