Q:

Find the coordinates of the vertices of the triangle and compute the area of the triangle using the distance formula. (round to the nearest integer in units2)

Accepted Solution

A:
We are asked to find two things, namely:Β 1. The coordinates of the vertices of the triangle.2. Compute the area of the triangle using the distance formula.Let's solve this exercise step by step, so:1. To do this we will see the the figure below to find out the points of each vertex. Thus, the three points are as follows:[tex]P_{1}(1, -5)[/tex][tex]P_{2}(4, -2)[/tex][tex]P_{3}(-3, 5)[/tex]2. The area of a triangle is given by this formula:[tex]A=\frac{bh}{2}[/tex]where b is the base and h the height of the triangle. We can use the Distance Formula to solve this problem so:[tex]b=\overline{P_{1}P_{2}}=\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2} \\ \therefore b=\sqrt{(1-4)^2+[-5-(-2)]^2}=3\sqrt{2}[/tex][tex]h=\overline{P_{2}P_{3}}=\sqrt{(x_{2}-x_{3})^2+(y_{2}-y_{3})^2} \\ \therefore b=\sqrt{[4-(-3)]^2+(-2-5)^2}=3\sqrt{2}=7\sqrt{2}[/tex]Finally the area is:[tex]A=\frac{3\sqrt{2}\times 7\sqrt{2}}{2} \rightarrow \boxed{A=21 \ units^2}[/tex]