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.