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 \(A(2,2)\), \(B(4,6)\), \(C(8,2)\)
• Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
• For \(AB\): \(d_{AB}=\sqrt{(4 - 2)^2+(6 - 2)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\)
• For \(BC\): \(d_{BC}=\sqrt{(8 - 4)^2+(2 - 6)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\)
• For \(AC\): \(d_{AC}=\sqrt{(8 - 2)^2+(2 - 2)^2}=\sqrt{36}=6\)

Step2: Calculate the perimeter \(P\)

\(P=d_{AB}+d_{BC}+d_{AC}=2\sqrt{5}+4\sqrt{2}+6\approx2\times2.24 + 4\times1.41+6=4.48+5.64 + 6=16.12\)

Step3: Calculate the area \(A\)

• Base \(b = AC = 6\), height \(h\) is the vertical distance from \(B\) to \(AC\) (which is \(6 - 2=4\))
• Use the formula \(A=\frac{1}{2}\times b\times h\)

\(A=\frac{1}{2}\times6\times4 = 12\)

Answer:

\(P\approx16.12\) units, \(A = 12\) units²