Question

find the perimeter and area of this figure.
p = ? units
a = units²

Explanation:

Step1: Find the lengths of the sides

Let the vertices be \(A(2,2)\), \(B(5,6)\), \(C(8,2)\).
Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
For \(AB\): \(d_{AB}=\sqrt{(5 - 2)^2+(6 - 2)^2}=\sqrt{9 + 16}=\sqrt{25}=5\)
For \(BC\): \(d_{BC}=\sqrt{(8 - 5)^2+(2 - 6)^2}=\sqrt{9+16}=\sqrt{25}=5\)
For \(AC\): \(d_{AC}=\sqrt{(8 - 2)^2+(2 - 2)^2}=\sqrt{36+0}=6\)

Step2: Calculate the perimeter

\(P=d_{AB}+d_{BC}+d_{AC}=5 + 5+6=16\)

Step3: Calculate the area

Use the formula for the area of a triangle \(A=\frac{1}{2}\times base\times height\). Here, base \(AC = 6\), height is the vertical distance from \(B\) to \(AC\) which is \(6 - 2=4\)
\(A=\frac{1}{2}\times6\times4 = 12\)

Answer:

\(P = 16\) units, \(A=12\) units\(^2\)