Question

what is the area of the triangle? square units

Explanation:

Step1: Find the base length

The two points \((-5, -2)\) and \((1, -2)\) have the same y - coordinate, so the distance between them (the base of the triangle) is calculated by the formula for the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) when \(y_1 = y_2\): \(d=\vert x_2 - x_1\vert\).
Here, \(x_1=-5\), \(x_2 = 1\), so the base \(b=\vert1-(-5)\vert=\vert1 + 5\vert=6\).

Step2: Find the height length

The height of the triangle is the vertical distance from the point \((-3,5)\) to the line containing the base (which is \(y=-2\)). The vertical distance between a point \((x,y)\) and the line \(y = k\) is \(\vert y - k\vert\).
Here, \(y = 5\) and \(k=-2\), so the height \(h=\vert5-(-2)\vert=\vert5 + 2\vert=7\).

Step3: Calculate the area of the triangle

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\).
Substituting \(b = 6\) and \(h=7\) into the formula, we get \(A=\frac{1}{2}\times6\times7\).
First, \(\frac{1}{2}\times6 = 3\), then \(3\times7=21\).

Answer:

21