Question

type the correct answer in each box. if necessary, round your answer(s) to the nearest hundredth. the vertices of △abc are a(-2, 2), b(6, 2), and c(0, 8). the perimeter of △abc is units, and its area is square units.

Explanation:

Step1: Calculate the length of AB

The coordinates of A are (-2, 2) and B are (6, 2). Since the y - coordinates are the same, the distance \(AB=\vert6 - (-2)\vert=\vert6 + 2\vert = 8\) units.

Step2: Calculate the length of AC

The coordinates of A are (-2, 2) and C are (0, 8). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(x_1=-2,y_1 = 2,x_2 = 0,y_2=8\).
\(AC=\sqrt{(0 - (-2))^2+(8 - 2)^2}=\sqrt{(2)^2+(6)^2}=\sqrt{4 + 36}=\sqrt{40}\approx6.32\) units.

Step3: Calculate the length of BC

The coordinates of B are (6, 2) and C are (0, 8). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(x_1 = 6,y_1=2,x_2=0,y_2 = 8\).
\(BC=\sqrt{(0 - 6)^2+(8 - 2)^2}=\sqrt{(-6)^2+(6)^2}=\sqrt{36+36}=\sqrt{72}\approx8.49\) units.

Step4: Calculate the perimeter

The perimeter \(P=AB + AC+BC\). Substituting the values: \(P = 8+\sqrt{40}+\sqrt{72}\approx8 + 6.32+8.49=22.81\) units.

Step5: Calculate the area

The base of the triangle can be taken as AB = 8 (since it is a horizontal line). The height is the vertical distance from point C to the line AB. The y - coordinate of AB is 2 and the y - coordinate of C is 8, so the height \(h=\vert8 - 2\vert=6\) units.
The area of a triangle \(A=\frac{1}{2}\times base\times height\). So \(A=\frac{1}{2}\times8\times6 = 24\) square units.

Answer:

The perimeter of \(\triangle ABC\) is \(\boldsymbol{22.81}\) units, and its area is \(\boldsymbol{24}\) square units.