Question

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

Explanation:

Step1: Identify the base and height for area

Let's assume the base of the triangle lies on the horizontal - axis. Counting the grid - squares, if each grid - square has a side - length of 1 unit, the base $b$ of the triangle is 6 units and the height $h$ is 7 units.

Step2: Calculate the area

The formula for the area of a triangle is $A=\frac{1}{2}bh$. Substitute $b = 6$ and $h = 7$ into the formula:
$A=\frac{1}{2}\times6\times7=21$ square units.

Step3: Use the distance formula to find the lengths of the non - horizontal sides

Let the vertices of the triangle be $(2,2)$, $(8,2)$ and $(5,9)$. The distance formula 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 the side between $(2,2)$ and $(5,9)$:
$d_1=\sqrt{(5 - 2)^2+(9 - 2)^2}=\sqrt{3^2+7^2}=\sqrt{9 + 49}=\sqrt{58}\approx7.62$ units.
For the side between $(8,2)$ and $(5,9)$:
$d_2=\sqrt{(5 - 8)^2+(9 - 2)^2}=\sqrt{(-3)^2+7^2}=\sqrt{9 + 49}=\sqrt{58}\approx7.62$ units.
The length of the base is $b = 6$ units.

Step4: Calculate the perimeter

The perimeter $P$ of the triangle is $P=d_1 + d_2+b$. Substitute $d_1\approx7.62$, $d_2\approx7.62$ and $b = 6$ into the formula:
$P\approx7.62+7.62 + 6=21.24$ units.

Answer:

$P = 21.24$ units, $A = 21$ units$^2$