Question

1. centroid of a triangle - use the back page for your equations and solutions.

triangle mpv is shown, where point r
is the centroid of the triangle.
mr = 5x - 9, rp = 3w + 7, rv = 12y - 9,
rw = 2x, rt = 4y + 2, and
hr = w + 9.
a. find the value of each variable.
x=
y=
w=

Explanation:

Step1: Use centroid property for \(x\)

The centroid divides the median in the ratio \(2:1\). For median \(MH\), \(MR = 2RW\). Given \(MR=5x - 9\) and \(RW = 2x\).
So, \(5x-9=2\times(2x)\)
\(5x - 9=4x\)
\(5x-4x=9\)
\(x = 9\)

Step2: Use centroid property for \(y\)

For median \(VT\), \(RV = 2RT\). Given \(RV=12y - 9\) and \(RT = 4y+2\)
So, \(12y-9 = 2\times(4y + 2)\)
\(12y-9=8y + 4\)
\(12y-8y=4 + 9\)
\(4y=13\)
\(y=\frac{13}{4}=3.25\)

Step3: Use centroid property for \(w\)

For median \(PV\), \(RP = 2HR\). Given \(RP=3w + 7\) and \(HR=w + 9\)
So, \(3w+7=2\times(w + 9)\)
\(3w+7=2w+18\)
\(3w-2w=18 - 7\)
\(w = 11\)

Answer:

\(x = 9\)
\(y = 3.25\)
\(w = 11\)