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Chapter 13: Q. 71 (page 1041)

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鈥�72, prove the given statements.

Use the integral definition for the centroid to show that the centroid of T is point P from Exercise 70.

Short Answer

Expert verified

The Center of mass of a triangle is The centroid of the triangle.

So centroid of the triangle is p(x,y)=a+c3,d3.

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. The centroid of the triangle.

The Center of mass of a triangle is The centroid of the triangle.

x coordinate of the center of mass is following.

x=鈭�鈭�惟xdA鈭�鈭�惟dAx=鈭�0c鈭�0dcxxdydx+鈭�0c鈭�0d/c鈭�a(x鈭�a)xdydx鈭�0c鈭�0d/cxdydxx=a+c3

y coordinate of the center of mass is following.

y=鈭�鈭�惟ydA鈭�鈭�惟dAy=鈭�0c鈭�0dcxydydx+鈭�0c鈭�0d/c鈭�a(x鈭�a)ydydx鈭�0c鈭�0d/cxdydxy=d3

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

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