Question

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

Explanation:

Step1: Label the vertices

Let the vertices be \(A(2,2)\), \(B(4,6)\), \(C(8,2)\)

Step2: Calculate side lengths

• \(AB\): Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(A(2,2)\) and \(B(4,6)\), \(AB=\sqrt{(4 - 2)^2+(6 - 2)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\approx4.47\)
• \(BC\): For \(B(4,6)\) and \(C(8,2)\), \(BC=\sqrt{(8 - 4)^2+(2 - 6)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\approx5.66\)
• \(AC\): For \(A(2,2)\) and \(C(8,2)\), \(AC=\sqrt{(8 - 2)^2+(2 - 2)^2}=6\)
• \(P=AB + BC+AC\approx4.47+5.66 + 6=16.13\approx16\) (using grid - counting approximation: count the lengths as \(AB = 5\), \(BC = 5\), \(AC = 6\), \(P=5 + 5+6 = 16\))

Step3: Calculate the area

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

Answer:

$P = 16$ units, $A = 12$ units²