Question

this question has two parts. first, answer part a. then, answer part b. part a 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. b =, h = √3, so a = 1/2( )(√3) = √3 units². part b c. find the area of the square that is not covered by the triangle. select an exact value and then round to the nearest tenth. justify your reasoning. the area not covered by the triangle is select choice the area of the square select choice the area of the triangle. so, a select choice (16 select choice √3) units² or about select choice units².

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 $s=\sqrt{(2-(-2))^2+(3 - 3)^2}=\sqrt{(4)^2+0^2}=4$.

Step2: Calculate area of square

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

Step3: Find base of triangle

The base of the equilateral triangle $EF$ with $E(0,\sqrt{3})$, $F(1,0)$ and $G(-1,0)$. The length of the base $b$ (distance between $G$ and $F$) is $b=|1-(-1)| = 2$.

Step4: Calculate area of triangle

The area formula of a triangle is $A=\frac{1}{2}bh$. Given $b = 2$ and $h=\sqrt{3}$, so $A=\frac{1}{2}(2)(\sqrt{3})=\sqrt{3}$ units$^2$.

Step5: Find area not covered by triangle

The area not covered by the triangle is the area of the square minus the area of the triangle. So $A_{not\ covered}=16-\sqrt{3}$.

Answer:

a. $s = 4$, so $A=4^2 = 16$ units$^2$
b. $b = 2$, $h=\sqrt{3}$, so $A=\frac{1}{2}(2)(\sqrt{3})=\sqrt{3}$ units$^2$
c. The area not covered by the triangle is the area of the square minus the area of the triangle. So $A = 16-\sqrt{3}$ units$^2$ or about $16 - 1.732=14.3$ units$^2$