Question

Triangle W, X, YWXY is formed by connecting the midpoints of the side of triangle T, U, VTUV. The lengths of the sides of triangle W, X, YWXY are shown. Find the perimeter of triangle T, U, VTUV. Figures not necessarily drawn to scale

Answer 1

To find the perimeter of triangle TUV, you can use the information given for triangle WXY. The triangles TUV and WXY are similar, and their corresponding sides are in proportion because connecting the midpoints of the sides of a triangle creates a smaller, similar triangle. This means that the ratio of the sides of triangle TUV to triangle WXY is the same.

Let's denote the sides of triangle TUV as T1U1, U1V1, and VT1, and the sides of triangle WXY as WX, XY, and YW. You are given the following side lengths for triangle WXY:

WX = 8

XY = 10

YW = 12

Now, find the ratios of the corresponding sides:

(T1U1 / WX) = (U1V1 / XY) = (VT1 / YW)

Substitute the values:

(T1U1 / 8) = (U1V1 / 10) = (VT1 / 12)

Now, you can find the lengths of the sides of triangle TUV:

T1U1 = (8/12) * 8 = 4/3 * 8 = 32/3

U1V1 = (10/12) * 10 = 5/6 * 10 = 50/6 = 25/3

VT1 = (12/12) * 12 = 12

Now, you have the side lengths of triangle TUV:

T1U1 = 32/3

U1V1 = 25/3

VT1 = 12

To find the perimeter, add these side lengths:

Perimeter (TUV) = T1U1 + U1V1 + VT1 = (32/3) + (25/3) + 12 = (32/3 + 25/3) + 12 = (57/3) + 12 = 19 + 12 = 31

So, the perimeter of triangle TUV is 31 units.

Explanation: