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: Determine side - length of square

The distance between \(D(-2,-1)\) and \(C(2,-1)\) is used to find the side - length of the square. 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, \(x_1=-2,y_1 = - 1,x_2=2,y_2=-1\), so \(s=\sqrt{(2-(-2))^2+(-1 - (-1))^2}=\sqrt{(4)^2+0^2}=4\).

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: Determine base of triangle

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

Step4: Calculate area of triangle

The area formula of a triangle is \(A=\frac{1}{2}bh\). Given \(b = 2\) and \(h=\sqrt{3}\), we substitute these values into the formula: \(A=\frac{1}{2}\times2\times\sqrt{3}=\sqrt{3}\) square units.

Answer:

a. The area of the square is 16 square units.
b. The area of the triangle is \(\sqrt{3}\) square units.