Question

a. find the area of the square. show your work.
s = 4, so a = 4^2 = 16 units^2
b. find the area of the triangle. show your work.
b =
h = \sqrt{3}, so a = \frac{1}{2}( )(\sqrt{3}) = \sqrt{3} units^2.

Explanation:

Step1: Calculate 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 $s=\sqrt{(2+2)^2+(3 - 3)^2}=4$.

Step2: Calculate area of square

The area formula of a square is $A = s^2$. Substituting $s = 4$ into the formula, we get $A=4^2=16$ square units.

Step3: Calculate base of triangle

The base of $\triangle EFG$ is the distance between $G(-1,0)$ and $F(1,0)$. Using the distance formula $b=\sqrt{(1 + 1)^2+(0-0)^2}=2$.

Step4: Calculate area of triangle

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

Answer:

a. The area of the square is 16 square units.
b. The base of the triangle $b = 2$, and the area of the triangle is $\sqrt{3}$ square units.