Question

find the perimeter and area of triangle abc if each unit on the graph measures 1 centimeter. round answers to the nearest tenth if necessary. perimeter cm area cm²

Explanation:

Step1: Calculate the lengths of the sides

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For side $AC$ with $A(0,4)$ and $C(-4,0)$:
$AC=\sqrt{( - 4-0)^2+(0 - 4)^2}=\sqrt{(-4)^2+( - 4)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\approx5.7$ cm.
For side $AB$ with $A(0,4)$ and $B(2,0)$:
$AB=\sqrt{(2 - 0)^2+(0 - 4)^2}=\sqrt{2^2+( - 4)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\approx4.5$ cm.
For side $BC$ with $B(2,0)$ and $C(-4,0)$:
$BC=\sqrt{( - 4-2)^2+(0 - 0)^2}=\sqrt{(-6)^2+0^2}=6$ cm.

Step2: Calculate the perimeter

Perimeter $P=AC + AB+BC$.
$P\approx5.7+4.5 + 6=16.2$ cm.

Step3: Calculate the area

The base of the triangle is the length of $BC$, $b = 6$ cm, and the height is the $y$-coordinate of point $A$, $h = 4$ cm.
Using the area formula for a triangle $A=\frac{1}{2}bh$.
$A=\frac{1}{2}\times6\times4 = 12$ $cm^2$.

Answer:

perimeter: $16.2$ cm
area: $12$ $cm^2$