Question

The following points are the vertices of a shaded triangle. (2,5), (6,−3), (−2,−3) a. Write a system of linear inequalities represented by the shaded triangle. Left boundary inequality: Right boundary inequality: Lower boundary inequality: b. Find the area of the triangle. square units

Answer 1

We are going to find the plot where this coordinates are in the graph.

Your graph will be in the attachment below.

~Jurgen

Answer 2

The given triangle ha an horizontal base from which the altitude can be

found from the distance to the opposite vertex.

The system of linear inequalities are;

• y ≤ -2·x + 9

• y ≤ 2·x + 1

• y ≥ -3

Please find attached the shaded triangle created with MS Excel

• The area of the triangle, A = 32 square units

Reasons:

The given vertices of the triangle are; (2, 5), (6, -3), and (-2, -3).

Let A, B, and C, represent the vertices, we have;

A(2, 5), B(6, -3), and C(-2, -3)

• $\displaystyle Slope \ of \ AB = \mathbf{ \frac{- 3 - 5}{6 - 2}} = \frac{-8}{4} = -2$

Equation of AB is; y - 5 = (-2)·(x - 2)

y = -2·x + 4 + 5 = -2·x + 9

y = -2·x + 9

$\displaystyle Slope \ of \ BC = \frac{- 3 - (-3)}{-2 - 6} = \frac{0}{4} = 0$

Equation of BC is; y - (-3) = 0

y = -3

• $\displaystyle Slope \ of \ AC = \mathbf{\frac{- 3 - 5}{-2 - 2}} = \frac{-8}{-4} = 2$

Equation of AC is y - 5 = 2·(x - 2)

y = 2·x - 4 + 5

y = 2·x + 1

By plotting the above equations, we have the attached graph, with the following system of inequalities;

y ≤ -2·x + 9

y ≤ 2·x + 1

y ≥ -3

From the graph, we have;

The base of the triangle is an horizontal line, y = x

The length of the base of the triangle is given by the intersection of the lines AB and AC at point B and C = BC = 6 - (-2) = 8

The height of the triangle = The vertical distance of vertex A from CB

Therefore;

The height of the triangle = 5 - (-3) = 8

Area of a triangle, A = Half base × Height

A = 0.5 × 8 × 8 = 32

The area of the triangle, A = 32 square units

Learn more about finding the area of a triangle on a graph here:

https://brainly.com/question/21735282