Question

triangle wny is shown, where points d, r, and b are midpoints of each side of the triangle. rb = 11, yn = (5x), dr = (\frac{1}{2}x + 12), nw = (5z + 1), and db = (3z - 2). what is the perimeter of \triangle wny?

Explanation:

Step1: Use the mid - segment theorem for \(RB\) and \(YN\)

The mid - segment theorem states that the length of a mid - segment (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side.
Since \(RB\) is a mid - segment and \(RB = 11\), and \(RB=\frac{1}{2}YN\).
We have \(YN = 2RB\). Substituting \(RB = 11\), we get \(YN=2\times11 = 22\).
Since \(YN = 5x\), then \(5x=22\), \(x=\frac{22}{5}\).

Step2: Use the mid - segment theorem for \(DR\) and \(WY\)

\(DR\) is a mid - segment. Let \(WY\) be the third side. \(DR=\frac{1}{2}WY\). But we don't need to use \(DR\) for the perimeter calculation.

Step3: Use the mid - segment theorem for \(DB\) and \(NW\)

Since \(DB\) is a mid - segment and \(DB=\frac{1}{2}NW\).
Given \(NW = 5z + 1\) and \(DB=3z - 2\).
By the mid - segment theorem \(5z + 1=2(3z - 2)\).
Expand the right - hand side: \(5z + 1 = 6z-4\).
Subtract \(5z\) from both sides: \(1=6z - 5z-4\).
Add \(4\) to both sides: \(z=5\).

Step4: Calculate the lengths of the sides of \(\triangle WNY\)

We know \(YN = 22\).
For \(NW\): substitute \(z = 5\) into \(NW=5z + 1\). Then \(NW=5\times5 + 1=26\).
For \(WY\): Since \(DB\) is a mid - segment and \(DB = 3z-2\), when \(z = 5\), \(DB=3\times5 - 2=13\). And \(WY = 2DB\), so \(WY = 26\).

Step5: Calculate the perimeter of \(\triangle WNY\)

The perimeter \(P\) of \(\triangle WNY\) is \(P=YN + NW+WY\).
Substitute \(YN = 22\), \(NW = 26\), \(WY = 26\).
\(P=22 + 26+26\).
\(P = 74\).

Answer:

The perimeter of \(\triangle WNY\) is \(74\).