Question

centroid mixed practice

1. if v is the centroid of ( \triangle pqr ), ( sr = 21 ), ( vu = 8 ) and ( pt = 15 ). find each measure.

( sv=)
( vr=)
( qv=)
( qu=)
( pv=)
( vt=)

1. find the centroid of ( (-2,6),(-2,-4) ), and ( (5,-2) )
2. find the centroid of ( \triangle abc ) and plot the point.
3. ( \triangle jkl ) has two vertices located at ( (-2,8) ) and ( (4,2) ). the centroid is located at ( (6,9) ) what is the third vertex of the triangle?
4. a triangle with vertices of ( (-3,-1) ) and ( (2,6) ) has a centroid of ( (1,\frac{2}{3}) ). what are the coordinates of the third vertex?

Explanation:

Step1: Use centroid property for \(SV\) and \(VR\)

The centroid divides the median in the ratio \(2:1\). For median \(QS\), \(SR = 21\). Since \(V\) is the centroid, \(SV=\frac{2}{3}SR\).
\(SV=\frac{2}{3}\times21 = 14\)
\(VR=\frac{1}{3}SR=\frac{1}{3}\times21 = 7\)

Step2: Use centroid property for \(QV\) and \(QU\)

For median \(QU\), \(VU = 8\). \(QV = 2VU\)
\(QV=2\times8 = 16\)
\(QU=QV + VU=16 + 8=24\)

Step3: Use centroid property for \(PV\) and \(VT\)

For median \(PT\), \(PT = 15\). \(PV=\frac{2}{3}PT\)
\(PV=\frac{2}{3}\times15 = 10\)
\(VT=\frac{1}{3}PT=\frac{1}{3}\times15 = 5\)

Answer:

\(SV = 14\), \(VR = 7\), \(QV = 16\), \(QU = 24\), \(PV = 10\), \(VT = 5\)