Question

1. name the segment parallel to the given segment.

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: Apply the mid - segment theorem for problem 2

The mid - segment theorem states that a segment connecting the midpoints of two sides of a triangle is parallel to the third side.

• a) In the first figure, since \(X\) and \(Z\) are mid - points (by the tick marks), \(\overline{YZ}\parallel\overline{XT}\)
• b) \(\overline{RS}\parallel\overline{XZ}\) (because \(X\) and \(Y\) are mid - points of \(RT\) and \(RS\) respectively, and using the mid - segment property in a more complex figure with multiple mid - segments)
• c) \(\overline{XY}\parallel\overline{SZ}\) (by the mid - segment theorem)

Step2: Apply the mid - segment theorem for problem 4

The mid - segment theorem also states that the length of the mid - segment is half the length of the third side.

• a) Since \(CT\) is a mid - segment parallel to \(EN\) (not directly used here), but \(PT\) is a mid - segment. If \(PT\) is a mid - segment, and \(PT = 13\), then \(AE=2PT\). So \(AE = 2\times13=26\)
• b) Since \(CP\) is a mid - segment, \(AN = 2CP\). Given \(CP = 29\), then \(AN=2\times29 = 58\)
• c) Since \(C\), \(P\), and \(T\) are mid - points, \(CT=\frac{1}{2}EN\). Given \(EN = 43\), then \(CT=\frac{43}{2}=21.5\)
• d) The perimeter of \(\triangle AEN\) is \(AE + EN+AN\). Substitute \(AE = 26\), \(EN = 43\), and \(AN = 58\). So the perimeter \(P=26 + 43+58=127\)

Answer:

2.
a) \(\overline{XT}\)
b) \(\overline{XZ}\)
c) \(\overline{SZ}\)
4.
a) \(26\)
b) \(58\)
c) \(21.5\)
d) \(127\)