Question

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

Explanation:

Step1: Identify the side - lengths using the distance formula

Let the vertices of the triangle be \(A(3,2)\), \(B(3,5)\), \(C(7,5)\).
The distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For side \(AB\) with \(x_1 = 3,y_1 = 2,x_2 = 3,y_2 = 5\), \(d_{AB}=\sqrt{(3 - 3)^2+(5 - 2)^2}=\sqrt{0 + 9}=3\).
For side \(BC\) with \(x_1 = 3,y_1 = 5,x_2 = 7,y_2 = 5\), \(d_{BC}=\sqrt{(7 - 3)^2+(5 - 5)^2}=\sqrt{16+0}=4\).
For side \(AC\) with \(x_1 = 3,y_1 = 2,x_2 = 7,y_2 = 5\), \(d_{AC}=\sqrt{(7 - 3)^2+(5 - 2)^2}=\sqrt{16 + 9}=\sqrt{25}=5\).

Step2: Calculate the perimeter

The perimeter \(P\) of a triangle is the sum of the lengths of its sides. So \(P=d_{AB}+d_{BC}+d_{AC}=3 + 4+5 = 12\).

Step3: Calculate the area

The area \(A\) of a right - triangle (since \(AB\) is vertical, \(BC\) is horizontal, and \(AB\perp BC\)) is \(A=\frac{1}{2}\times base\times height\). Here, the base \(BC = 4\) and the height \(AB = 3\). So \(A=\frac{1}{2}\times4\times3=6\).

Answer:

\(P = 12\) units, \(A = 6\) units²