Question

triangle xyz is formed by connecting the midpoints of the side of triangle uvw. the lengths of the sides of triangle uvw are shown. find the perimeter of triangle xyz. figures not necessarily drawn to scale.

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of a mid - segment of a triangle (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side.
Let the sides of \(\triangle UVW\) be \(a = 10\), \(b = 10\), and \(c = 8\).
If \(XY\), \(YZ\), and \(ZX\) are mid - segments of \(\triangle UVW\), then \(XY=\frac{1}{2}UW\), \(YZ = \frac{1}{2}UV\), and \(ZX=\frac{1}{2}VW\).

Step2: Calculate the lengths of the sides of \(\triangle XYZ\)

\(XY=\frac{1}{2}\times8 = 4\), \(YZ=\frac{1}{2}\times10 = 5\), \(ZX=\frac{1}{2}\times10 = 5\)

Step3: Calculate the perimeter of \(\triangle XYZ\)

The perimeter \(P\) of a triangle is \(P=XY + YZ+ZX\).
\(P=4 + 5+5\)

Answer:

\(14\)