Question

find the perimeter and area of this figure.
round your answer to the nearest hundredth

Explanation:

Step1: Label the points

Let the vertices be \(A(2,2)\), \(B(5,9)\), \(C(8,2)\)

Step2: Calculate side lengths

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

$$\begin{align*} AB&=\sqrt{(5 - 2)^2+(9 - 2)^2}\\ &=\sqrt{3^2+7^2}\\ &=\sqrt{9 + 49}\\ &=\sqrt{58}\approx7.62 \end{align*}$$

For \(BC\):

$$\begin{align*} BC&=\sqrt{(8 - 5)^2+(2 - 9)^2}\\ &=\sqrt{3^2+(-7)^2}\\ &=\sqrt{9 + 49}\\ &=\sqrt{58}\approx7.62 \end{align*}$$

For \(AC\):

$$\begin{align*} AC&=\sqrt{(8 - 2)^2+(2 - 2)^2}\\ &=\sqrt{6^2+0^2}\\ &=6 \end{align*}$$

Step3: Calculate perimeter \(P\)

\(P=AB + BC+AC\approx7.62+7.62 + 6=20.49\)

Step4: Calculate area \(A\)

Base \(b = AC = 6\), height \(h\) (vertical distance from \(B\) to \(AC\)) is \(9 - 2=7\)
Using the formula \(A=\frac{1}{2}\times b\times h\)
\(A=\frac{1}{2}\times6\times6 = 18\) (since the base \(AC = 6\) (from \(x = 2\) to \(x = 8\)) and height is \(6\) (from \(y=2\) to \(y = 8\) in the vertical direction relative to the base))

Answer:

$P = 20.49$ units, $A = 18$ units²