Question

6.3. more triangle theorems
1.
if triangle ( abc ) is an equilateral triangle with sides of 12 feet, and points ( x ), ( y ), and ( z ) are the midpoints of each side, find the perimeter of triangle ( xyz ).

Explanation:

Step1: Recall Midsegment Theorem

The midsegment of a triangle is parallel to the third side and half as long. For triangle \(ABC\), \(X\), \(Y\), \(Z\) are midpoints, so \(XY\), \(YZ\), \(ZX\) are midsegments.

Step2: Find length of one midsegment

Each side of \(ABC\) is \(12\) ft. A midsegment length is \(\frac{1}{2}\times12 = 6\) ft.

Step3: Calculate perimeter of \(XYZ\)

Triangle \(XYZ\) has three sides, each \(6\) ft. Perimeter \(= 6 + 6 + 6 = 18\) ft.

Answer:

18 feet