The points (1,2), (5,5),(1,6) are the vertices of a shaded triangle.

a. Write a system of linear inequalities represented by the shaded triangle.
b. Find the area of the triangle.

Just give us a damn answer

a. Well, to determine the linear inequalities represented by the shaded triangle, we can start by finding the equations of the lines formed by connecting the given points.

First, let's find the equation of the line passing through (1,2) and (5,5). The slope of this line is (5 - 2)/(5 - 1) = 3/4. Using the point-slope form of a line, we have: y - 2 = (3/4)(x - 1). Simplifying this equation, we get: y - 2 = (3/4)x - 3/4.

Similarly, let's find the equation of the line passing through (5,5) and (1,6). The slope of this line is (6 - 5)/(1 - 5) = -1/4. Using the point-slope form of a line, we have: y - 5 = (-1/4)(x - 5). Simplifying this equation, we get: y - 5 = (-1/4)x + 5/4.

Now, let's find the equation of the line passing through (1,6) and (1,2), which is simply x = 1.

So, the system of linear inequalities represented by the shaded triangle is:
y - 2 ≥ (3/4)x - 3/4, (Equation 1)
y - 5 ≤ (-1/4)x + 5/4, (Equation 2)
x = 1. (Equation 3)

Note: The inequality signs are chosen based on the side of the line where we want the shaded region to be.

b. To find the area of the triangle, we can use the formula for the area of a triangle given its vertices.

Using the formula: Area = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Substituting the coordinates of the vertices, we have:

Area = 1/2 |1(5 - 6) + 5(6 - 2) + 1(2 - 5)| = 1/2 |-1 + 20 - 3| = 1/2 |16| = 8

So, the area of the triangle is 8.

To write a system of linear inequalities represented by the shaded triangle, we can use the concept of slope-intercept form.

Step 1: Determine the equation of each of the three sides of the triangle.

Side 1: (1,2) to (5,5)
The slope of this line is (5-2)/(5-1) = 3/4.
Using the point-slope form of a linear equation, we have: y - 2 = (3/4)(x - 1).
Rewriting it in slope-intercept form, we get: y = (3/4)x + 5/4.

Side 2: (5,5) to (1,6)
The slope of this line is (6-5)/(1-5) = -1/4.
Using the point-slope form of a linear equation, we have: y - 5 = (-1/4)(x - 5).
Rewriting it in slope-intercept form, we get: y = (-1/4)x + 29/4.

Side 3: (1,6) to (1,2)
This side is a vertical line and it can be expressed as x = 1.

Step 2: Convert the equations into inequalities based on the shaded region.

For Side 1, since the shaded region is below the line, the inequality is y ≤ (3/4)x + 5/4.

For Side 2, since the shaded region is below the line, the inequality is y ≤ (-1/4)x + 29/4.

For Side 3, since the shaded region is to the left of the line, the inequality is x ≥ 1.

Therefore, the system of linear inequalities represented by the shaded triangle is:
y ≤ (3/4)x + 5/4,
y ≤ (-1/4)x + 29/4,
x ≥ 1.

Now, let's move on to finding the area of the triangle.

Step 1: Calculate the base of the triangle.
The base of the triangle is the horizontal distance between the two points (5,5) and (1,6).
Using the distance formula, we have: base = sqrt((5-1)^2 + (5-6)^2) = sqrt(16+1) = sqrt(17).

Step 2: Calculate the height of the triangle.
The height of the triangle is the vertical distance between the side with a slope of 3/4 and the line x = 1.
Using the equation of the side with a slope of 3/4 (y = (3/4)x + 5/4), we can find the height by finding the y-coordinate where x = 1.
Substituting x = 1 in the equation gives us: y = (3/4)(1) + 5/4 = 8/4 = 2.

Step 3: Calculate the area of the triangle using the formula: area = (1/2) * base * height.
Plugging the values we obtained, we have: area = (1/2) * sqrt(17) * 2 = sqrt(17).

Therefore, the area of the triangle is sqrt(17) square units.

5) The following points are the vertices of a triangle: (0, –1), (4, 5), (4, –1).

a. Write a system of linear inequalities that represents the points inside the triangle.
b. Find the area of the triangle.

Y>3x

well, first, draw the triangle!

it is easy to find the area, since the base is a vertical line.
Now it should be clear that one inequality is
x > 1
Now determine the equations of the other two sides, and decide whether y is above or below each line.