Question

the coordinate grid shows an equilateral triangle that fits inside a square.
a. find the area of the square. show your work.
s =

, so a =

²=

units²
b. find the area of the triangle. show your work.

Explanation:

Step1: Find side - length of square

The distance between two adjacent vertices of the square, for example, between $A(-2,3)$ and $B(2,3)$. Using the distance formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ which is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $x_1=-2,y_1 = 3,x_2=2,y_2 = 3$. So $s=\sqrt{(2 - (-2))^2+(3 - 3)^2}=\sqrt{(2 + 2)^2+0^2}=4$.

Step2: Calculate area of square

The area formula of a square is $A=s^2$. Since $s = 4$, then $A = 4^2=16$ square units.

Step3: Find base and height of triangle

The base of the equilateral triangle $GF$ has length $b=|1-(-1)| = 2$. The height of the equilateral triangle from $E(0,\sqrt{3})$ to the $x -$axis is $h=\sqrt{3}$.

Step4: Calculate area of triangle

The area formula of a triangle is $A=\frac{1}{2}bh$. Substituting $b = 2$ and $h=\sqrt{3}$, we get $A=\frac{1}{2}\times2\times\sqrt{3}=\sqrt{3}$ square units.

Answer:

a. $s = 4$, so $A=4^2 = 16$ units$^2$
b. $A=\sqrt{3}$ units$^2$