Question

1. if c, p, and t are the midpoints of the sides of \\( \triangle a e n, p t = 13, e n = 43 \\), and \\( c p = 29 \\), find each measure.

a) \\( a e = \\)
b) \\( a n = \\)
c) \\( c t = \\)
d) perimeter of \\( \triangle a e n \\):

Explanation:

Step1: Use the mid - segment theorem for \(AE\)

The mid - segment theorem states that the length of a mid - segment (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side.
Since \(CP\) is a mid - segment parallel to \(AE\), \(CP=\frac{1}{2}AE\).
Given \(CP = 29\), then \(AE=2\times CP\).
\(AE = 2\times29=58\)

Step2: Use the mid - segment theorem for \(AN\)

Since \(PT\) is a mid - segment parallel to \(AE\), \(PT=\frac{1}{2}AE\) (already used for \(AE\)). Since \(CP\) is a mid - segment parallel to \(AE\) and \(CT\) is a mid - segment.
Since \(CP\) is a mid - segment of \(\triangle AEN\) (connecting mid - points of \(EA\) and \(EN\)), and \(PT\) is a mid - segment (connecting mid - points of \(EN\) and \(AN\)).
Since \(PT\) is a mid - segment, \(PT=\frac{1}{2}AE\) (not relevant here). For \(AN\), since \(CP\) is a mid - segment parallel to \(AE\) and \(CT\) is a mid - segment.
Since \(PT\) is a mid - segment of \(\triangle AEN\) (connecting mid - points of \(EN\) and \(AN\)), we know that \(AN = 2\times PT\)
Given \(PT=13\), then \(AN=2\times13 = 26\)

Step3: Use the mid - segment theorem for \(CT\)

Since \(CT\) is a mid - segment parallel to \(EN\), \(CT=\frac{1}{2}EN\)
Given \(EN = 43\), then \(CT=\frac{1}{2}\times43 = 21.5\)

Step4: Calculate the perimeter of \(\triangle AEN\)

The perimeter of \(\triangle AEN\) is \(P=AE + EN+AN\)
We found \(AE = 58\), \(EN = 43\), \(AN=26\)
\(P=58 + 43+26=127\)

Answer:

a) \(AE = 58\)
b) \(AN = 26\)
c) \(CT=21.5\)
d) Perimeter of \(\triangle AEN:127\)