Question

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

Explanation:

Step1: Calculate the lengths of the sides using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

• For \(AB\): \(A(2,8)\), \(B(16,2)\)

\(AB=\sqrt{(16 - 2)^2+(2 - 8)^2}=\sqrt{14^2+(- 6)^2}=\sqrt{196 + 36}=\sqrt{232}\approx15.23\)

• For \(BC\): \(B(16,2)\), \(C(6,2)\)

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

• For \(AC\): \(A(2,8)\), \(C(6,2)\)

\(AC=\sqrt{(6 - 2)^2+(2 - 8)^2}=\sqrt{4^2+(-6)^2}=\sqrt{16 + 36}=\sqrt{52}\approx7.21\)

Step2: Calculate the perimeter \(P=AB + BC+AC\)

\(P\approx15.23+10 + 7.21=32.44\)

Step3: Calculate the area using the formula for the area of a triangle with base \(BC\) and height (vertical distance from \(A\) to \(BC\))

The base \(BC = 10\), the height \(h\) is the difference in the \(y\) - coordinates of \(A\) and the line \(y = 2\) (since \(B\) and \(C\) have \(y = 2\)), so \(h=8 - 2=6\)
The area \(A=\frac{1}{2}\times base\times height=\frac{1}{2}\times10\times6 = 30\)

Answer:

The perimeter is \(32.44\) units and the area is \(30\) square units.