Question

type the correct answer in each box. if necessary, round your answers to the nearest hundredth. the vertices of δabc are a(2, 8), b(16, 2), and c(6, 2). the perimeter of δabc is units, and its area is square units.

Explanation:

Step1: Calculate lengths of sides

• For \( AB \): Use distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

\( A(2,8) \), \( B(16,2) \):
\( AB = \sqrt{(16 - 2)^2 + (2 - 8)^2} = \sqrt{14^2 + (-6)^2} = \sqrt{196 + 36} = \sqrt{232} \approx 15.23 \).

• For \( BC \): \( B(16,2) \), \( C(6,2) \):

\( BC = \sqrt{(6 - 16)^2 + (2 - 2)^2} = \sqrt{(-10)^2 + 0^2} = 10 \).

• For \( AC \): \( A(2,8) \), \( C(6,2) \):

\( AC = \sqrt{(6 - 2)^2 + (2 - 8)^2} = \sqrt{4^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.26 \).

Step2: Calculate perimeter

Perimeter \( = AB + BC + AC \approx 15.23 + 10 + 7.26 = 32.49 \).

Step3: Calculate area

Notice \( BC \) is horizontal (same \( y \)-coordinate), length \( 10 \). Height is vertical distance from \( A \) to \( BC \): \( 8 - 2 = 6 \).
Area \( = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 6 = 30 \).

Answer:

Perimeter: 32.49, Area: 30