Aquadrilateral has vertices at a(-5,5), b(1,8), c(4,2), and d(-2,-2). use slope to determine if the quadrilateral is a rectangle. show your work.

The quadrilateral is not a rectangle

Step-by-step explanation:

we know that

If a quadrilateral ABCD is a rectangle

then

Opposite sides are congruent and parallel and adjacent sides are perpendicular

Remember that

If two lines are parallel, then their slopes are the same

If two lines are perpendicular, then their slopes are opposite reciprocal (the product of their slopes is equal to -1)

The formula to calculate the slope between two points is equal to

we have

A(-5,5), B(1,8), C(4,2), and D(-2,-2)

Plot the figure to better understand the problem

see the attached figure

Find the slope of the four sides and then compare

step 1

Find slope AB

A(-5,5), B(1,8)

substitute in the formula

step 2

Find slope BC

B(1,8), C(4,2)

substitute in the formula

step 3

Find slope CD

C(4,2), and D(-2,-2)

substitute in the formula

step 4

Find slope AD

A(-5,5), D(-2,-2)

substitute in the formula

step 5

Verify if the opposites are parallel

Remember that

If two lines are parallel, then their slopes are the same

The opposite sides are

AB and CD

BC and AD

we have

so

It is not necessary to continue verifying, because two of the opposite sides are not parallel

therefore

The quadrilateral is not a rectangle

The problem seems to be lacking a question. But looking for the same problem from another source, we're looking for the new mode after adding the 6 numbers into the data set. 

Mode, in its most basic definition, is the number or data that is most repeated in a data set. For example, in {1, 1, 1, 2, 2, 3}, the most repeated number is 1. Hence, the mode is 1. 

Now, going back to Lucia's problem. Prior to adding the 6 numbers, the mode of the set of 87 numbers is already 31. That means 31 is the most repeated number in the set. Checking the numbers that were added, since another count for 31 is to be added, even if the number of 23, 26, 28, and 40 are increased by 1. 31 will still be the most repeated number in the set. Hence, the mode is still 31.

31