Question

find the perimeter and area of this figure.
p = ? units
a = units²
round your answer to the nearest hundredth

Explanation:

Step1: Find the lengths of the sides

Let the points be \(A(3,3)\), \(B(3,7)\), \(C(7,3)\).
The vertical side \(AB\): Using the distance formula \(d=\vert y_2 - y_1\vert\), \(AB=\vert7 - 3\vert=4\).
The horizontal side \(AC\): Using the distance formula \(d=\vert x_2 - x_1\vert\), \(AC=\vert7 - 3\vert=4\).
The hypotenuse \(BC\): Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(BC=\sqrt{(7 - 3)^2+(3 - 7)^2}=\sqrt{16 + 16}=\sqrt{32}\approx5.66\).

Step2: Calculate the perimeter

Perimeter \(P=AB + AC+BC\).
\(P = 4+4 + 5.66=16.89\).

Step3: Calculate the area

Area of a right - triangle \(A=\frac{1}{2}\times base\times height\).
Here, base \(AC = 4\), height \(AB = 4.25\) (wait, no, actually base \(AC = 4\), height \(AB = 3\) (from coordinates, vertical side from \(y = 3\) to \(y = 7\) is \(4\), horizontal from \(x = 3\) to \(x = 7\) is \(4\), no, wait the correct formula: for a right - triangle with legs \(a\) and \(b\), \(A=\frac{1}{2}ab\). Here \(a = 4\) (vertical side \(AB\)), \(b= 3\) (horizontal side \(AC\) from \(x = 3\) to \(x = 7\) is \(4\), no, wait the points: \(A(3,3)\), \(B(3,7)\) (so length \(AB=4\)), \(A(3,3)\), \(C(7,3)\) (length \(AC = 4\)). \(A=\frac{1}{2}\times4\times 3=10.5\).

Answer:

$P = 16.89$ units, $A = 10.5$ units²