Question

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

Explanation:

Step1: Find the lengths of the sides

Let the points be \(A(3,2)\), \(B(3,5)\), \(C(7,5)\).
The length of \(AB\): Using the distance formula for vertical line \(d=\vert y_2 - y_1\vert\), \(AB=\vert5 - 2\vert=3\).
The length of \(BC\): Using the distance formula for horizontal line \(d=\vert x_2 - x_1\vert\), \(BC=\vert7 - 3\vert = 4\).
The length of \(AC\): Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(AC=\sqrt{(7 - 3)^2+(5 - 2)^2}=\sqrt{16 + 9}=\sqrt{25}=5\).

Step2: Calculate the perimeter

Perimeter \(P=AB + BC+AC\).
\(P=3 + 4+5=12\).

Step3: Calculate the area

The figure is a right - triangle (since \(AB\perp BC\)). Area formula \(A=\frac{1}{2}\times base\times height\).
Here base \(BC = 4\) and height \(AB=3\).
\(A=\frac{1}{2}\times4\times3=6\).

Answer:

\(P = 12\) units, \(A=6\) units²