Question

extra credit #3: triangle midsegment

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 m

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

Explanation:

Step1: Recall the triangle midsegment theorem

The midsegment of a triangle is parallel to the third side and half its length.

Step2: Find \(AE\)

Since \(CP\) is a mid - segment parallel to \(EN\). By the mid - segment theorem, \(AE = 2CP\). Given \(CP=29\), then \(AE = 2\times29=58\).

Step3: Find \(AN\)

Since \(PT\) is a mid - segment parallel to \(AE\). By the mid - segment theorem, \(AN = 2PT\). Given \(PT = 13\), then \(AN=2\times13 = 26\).

Step4: Find \(CT\)

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

Step5: Find the perimeter of \(\triangle AEN\)

The perimeter of \(\triangle AEN\) is \(P=AE + EN+AN\). Substitute \(AE = 58\), \(EN = 43\), and \(AN = 26\) into the formula. Then \(P=58 + 43+26=127\).

Answer:

a. \(58\)
b. \(26\)
c. \(21.5\)
d. \(127\)