Question

triangle prg is shown, where points m, v, and n are the midpoints of each side of the triangle, and the perimeter of \\( \triangle prg \\) is 89. \\( gm=(4x - 1) \\), \\( gr=(8x + 3) \\), and \\( vr=(\frac{4}{3}x + 2) \\). what is the value of x?

Explanation:

Step1: Use the mid - point property

Since \(M\) is the mid - point of \(PG\), \(GM=\frac{1}{2}PG\). But we are not using this for the equation. Since \(V\) is the mid - point of \(PR\), by the mid - segment theorem (a line segment joining the mid - points of two sides of a triangle is parallel to the third side and half its length), we know that \(GM = VR\) (because \(MV\parallel GR\) and \(MV=\frac{1}{2}GR\), but here we use the fact that \(GM\) and \(VR\) are related as \(GM = VR\) from the mid - point and parallel line relationships in the triangle).
So, we set up the equation \(4x−1=\frac{4}{3}x + 2\).

Step2: Solve the linear equation

Subtract \(\frac{4}{3}x\) from both sides:
\(4x-\frac{4}{3}x-1=\frac{4}{3}x-\frac{4}{3}x + 2\)
\(\frac{12x - 4x}{3}-1=2\)
\(\frac{8x}{3}-1 = 2\)
Add \(1\) to both sides:
\(\frac{8x}{3}-1 + 1=2 + 1\)
\(\frac{8x}{3}=3\)
Multiply both sides by \(\frac{3}{8}\):
\(x=\frac{9}{8}=1.125\)

Answer:

\(x = \frac{9}{8}\)