Question

the coordinate grid shows an equilateral triangle that fits inside a square.
a(-2, 3) b(2, 3)
e(0, √3)
g(-1, 0) f(1, 0)
d(-2, -1) c(2, -1)
a. find the area of the square. show your work.
s =
, so a =
^2=
units^2
b. find the area of the triangle. show your work.
b =
, h = √3, so a = 1/2(
)(√3)=√3 units^2.

Explanation:

Step1: Find side - length of square

To find the side - length of the square, we use the distance formula for two points on the same horizontal or vertical line. For points \(A(-2,3)\) and \(B(2,3)\) (which are on the same horizontal line), the distance \(s\) between them is \(s=\vert x_2 - x_1\vert=\vert2-(-2)\vert = 4\).

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: Find base of triangle

For the equilateral triangle with vertices \(G(-1,0)\), \(F(1,0)\) and \(E(0,\sqrt{3})\), the base \(b\) of the triangle (the distance between \(G\) and \(F\)) is \(b=\vert x_2 - x_1\vert=\vert1 - (-1)\vert=2\).

Step4: Calculate area of triangle

The area formula of a triangle is \(A=\frac{1}{2}bh\). Given \(b = 2\) and \(h=\sqrt{3}\), we substitute these values into the formula: \(A=\frac{1}{2}(2)(\sqrt{3})=\sqrt{3}\) square units.

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\(^2\)