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Recall that a median of a triangle is a segment connecting a vertex of a triangle to the midpoint of the opposite side. Let T be the triangle with vertices (0,0),(a,0),and(c,d).In Exercises 70–72, prove the given statements.

Prove that the centroid of triangle T is two-thirds of the way from each vertex to the opposite side.

Short Answer

Expert verified

the centroid of triangle T is two-thirds of the way from each vertex to the opposite side.

APAX=2ca32+2d32ca22+d2APAX=2ca32+2d3232(2ca3)2+(2d3)2APAX=23

Step by step solution

01

Step 1. Given information.  

Vertices of given triangle T are(0,0),(a,0),and(c,d).

02

Step 2. Diagram of Triangle

Consider a triangle T of vertices B(0,0),C(a,0),andA(c,d).

The midpoint of AB is Zc2,d2.

The midpoint ofACisYa+c2,d2.

The midpoint of BCisXa2,d.

Medians of the triangle are AX, BY,and CZ.

Centroid isp(x,y)=a+c3,d3.

03

Step 3. Distance between centroid and vertex

Determine the distance between vertex A and centroid P.

AP=c-a+c32+d-d32AP=2ca32+2d32

Distance between vertex A and midpoint X.

AX=ca22+(d-0)2AX=ca22+d2

The ratio of AP to AX.

role="math" localid="1650437900251" APAX=2ca32+2d32ca22+d2APAX=2ca32+2d3232(2ca3)2+(2d3)2APAX=23

So centroid of triangle T is two-thirds of the way from each vertex to the opposite side.

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