Question

1. if c, p, and t are the mid - points of the sides of △aen, pt = 13, en = 43, and cp = 29, find each measure.

a) ae=
b) an=
c) ct=
d) perimeter of △aen:

Explanation:

Step1: Apply mid - point theorem

In a triangle, the line segment joining the mid - points of two sides is half the length of the third side. Since \(P\) and \(T\) are mid - points of \(EN\) and \(AN\) respectively, \(PT=\frac{1}{2}AE\). Given \(PT = 13\), then \(AE = 2\times PT\).
\[AE=2\times13 = 26\]

Step2: Find \(AN\)

Since \(C\) and \(P\) are mid - points of \(AE\) and \(EN\) respectively, \(CP=\frac{1}{2}AN\). Given \(CP = 29\), then \(AN=2\times CP\).
\[AN = 2\times29=58\]

Step3: Find \(CT\)

Since \(C\) and \(T\) are mid - points of \(AE\) and \(AN\) respectively, \(CT=\frac{1}{2}EN\). Given \(EN = 43\), then \(CT=\frac{1}{2}\times43 = 21.5\)

Step4: Find perimeter of \(\triangle AEN\)

The perimeter of \(\triangle AEN\) is \(P=AE + EN+AN\). We know \(AE = 26\), \(EN = 43\) and \(AN = 58\).
\[P=26 + 43+58=127\]

Answer:

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