1. Show that ABCD is a trapezoid.

(A: (-2, 5); B: (4, 5); C: (6,2); D: (2, -1)).

2. Write an equation in slope-intercept form for the line that passes through (5,0) and is perpendicular to the line given by y = -5/2x + 6.

2)

m = 2/5

y = mx + b

0 = (2/5)(5) + b

0 = 2 + b
b = -2

y = 2x/5 -2

in a trapezoid, one pair of opposite sides are parallel

Make a sketch

slope BC = (5-2)/(4-6) = 3/-2 = - 3/2
slope AD = (5-(-1))/)-2-2) = 6/-4 = -3/2

Yup, it is

I didn't check for the other pair
I could see that AB is horizontal, since both points have the same y value, which was not true for DC

How do I show that ABCD is a trapezoid, then?

AB = (5-5) /(4-(-2) = 0/6 = 0

AB is horizontal

BC = (2-5)/(6-4) = -3/2

CD = (-1-2)/(6-2) = -3/4

CD is not true

AD = (-1-5)/(2-(-2) = -6/4 = -3/2

So one pairs

To show that ABCD is a trapezoid, we need to prove that it has one pair of parallel sides.

1. Calculate the slopes of the lines formed by segments AB and CD using the formula:
slope = (y2 - y1) / (x2 - x1)

Slope of AB: m1 = (5 - 5) / (4 - (-2)) = 0 / 6 = 0
Slope of CD: m2 = (-1 - 2) / (2 - 6) = -3 / -4 = 3/4

2. If the slopes of two lines are equal, they are parallel. Since m1 = 0 and m2 = 3/4, AB and CD are not parallel.

3. Therefore, ABCD is not a trapezoid.

For the second question, we need to find the slope of a line perpendicular to y = -5/2x + 6 and then use the point-slope form to find its equation in slope-intercept form.

1. Determine the slope of the given line. In the equation y = -5/2x + 6, the coefficient of x is -5/2.

2. The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line. So, the slope of the line we want to find is the negative reciprocal of -5/2, which is 2/5.

3. Use the point-slope form to find the equation of the line. We have the point (5,0) and the slope m = 2/5.

y - y1 = m(x - x1)
y - 0 = (2/5)(x - 5)
y = (2/5)x - (2/5)*5
y = (2/5)x - 2

Therefore, the equation of the line passing through (5,0) and perpendicular to y = -5/2x + 6 is y = (2/5)x - 2 in slope-intercept form.