Question

if e is the midpoint of \\(\overline{ab}\\) and \\(ab = 30\\), what is \\(ae\\)?
\\(\bigcirc\\) a \\(\\ 5\\)
\\(\bigcirc\\) b \\(\\ 15\\)
\\(\bigcirc\\) c \\(\\ 30\\)
\\(\bigcirc\\) d \\(\\ 60\\)

question 8 (1 point)
(01.06 mc)
line segment with points l, p, r
if \\(lr = 12\\) and \\(lp = 7\\), find \\(pr\\). explain.
\\(\bigcirc\\) a \\(\\ pr = 19\\) because \\(12 + 7 = 19\\) according to the addition property of equality.
\\(\bigcirc\\) b \\(\\ pr = 5\\) because \\(12 - 7 = 5\\) according to the subtraction property of equality.
\\(\bigcirc\\) c \\(\\ pr = 19\\) because \\(lr + lp = pr\\) according to the segment addition postulate, and \\(12 + 7 = 19\\) using substitution.
\\(\bigcirc\\) d \\(\\ pr = 5\\) because \\(lp + pr = lr\\) according to the segment addition postulate, and \\(7 + 5 = 12\\) using substitution.

Explanation:

First Question (AE length):

Step1: Recall midpoint definition

A midpoint divides a segment into two equal parts, so \( AE = EB \) and \( AE + EB = AB \).

Step2: Substitute and solve

Since \( AE = EB \), let \( AE = x \), then \( x + x = AB \). Given \( AB = 30 \), so \( 2x = 30 \), \( x=\frac{30}{2}=15 \).

The Segment Addition Postulate states that if \( P \) is between \( L \) and \( R \), then \( LP + PR = LR \). We know \( LR = 12 \) and \( LP = 7 \). Substitute into the postulate: \( 7 + PR = 12 \). Solve for \( PR \): \( PR = 12 - 7 = 5 \), which matches option d's reasoning (using the postulate and substitution).

Answer:

b. 15

Question 8 (PR length):