When I tried answering a problem then looked at the answer in the back of my book, I was confused how to get to the correct answer. The problem is there is a trapezoid on a coordinate plane that is centered at the origin. The coordinates of the points have variables: T(-a,0), R(-b,c),A(b,c), P(a,0). The non parallel sides are TR and PA. I have to use the distance formula to find the length of segment TR and segment PA. In the selected answers sections of the book it says the answer is Yes;TR=PA= square root of a^2-2ab+b^2+c^2. How do I get this answer? The answer I got as the square root of a^2+b^2+c^2.

let's do PA first

PA = √( (a-b)^2 + (0-c)^2 )
= √( a^2 - 2ab + b^2 + c^2 ) , that was easy

now ..
TR = √( (-b + a)^2 + (c-0)^2 )
= √(b^2 - 2ab + a^2 + c^2 )
= PA

I don't see why you didn't get that, all I did was apply the distance formula

I think your problem was in expanding (a-b)^2
= (a-b)(a-b)= a^2 - ab - ab + b^2
= a^2 - 2ab + b^2

To find the length of segment TR and segment PA, you need to apply the distance formula. The distance formula calculates the distance between two points in a coordinate plane.

The formula for finding the distance between two points (x1, y1) and (x2, y2) is:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Let's apply this formula to find the length of segment TR:

Point T: T(-a, 0)
Point R: R(-b, c)

Substituting these values into the distance formula:

d(TR) = √((-b - (-a))^2 + (c - 0)^2)
= √((a - b)^2 + c^2)
= √(a^2 - 2ab + b^2 + c^2)

Similarly, for segment PA:

Point P: P(a, 0)
Point A: A(b, c)

d(PA) = √((b - a)^2 + c^2)
= √(a^2 - 2ab + b^2 + c^2)

These two answers, √(a^2 - 2ab + b^2 + c^2), are indeed equal. The book's answer combines the terms a^2, -2ab, and b^2 into a perfect square, (a - b)^2. Therefore, both TR and PA have the same length:

TR = PA = √(a^2 - 2ab + b^2 + c^2)

It seems that you missed the -2ab term when calculating the length. Make sure to double-check your calculations and take all the terms into account to arrive at the correct answer.