Question on "equation of a line". Read the question below.A(1,4), B(3,2) and C(7,5) are vertices of a triangle ABC. Find: 1. the coordinates of the centroid of triangle ABC 2. the equation of a...

Question on "equation of a line". Read the question below.

A(1,4), B(3,2) and C(7,5) are vertices of a triangle ABC. Find:

1. the coordinates of the centroid of triangle ABC

2. the equation of a line, through the centroid and parallel to AB.

Please provide steps.

Answer for 1. is (11/3,11/3)

Answer for 2. is 3x+3y=22

Asked on by novajose

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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The centroid of a triangle is the intercepting point of the medians of triangle.

The median of a triangle joins the vertex to the midpoint of the opposite side.

Therefore, we'll have to determine the equations of two medians of triangle and then to solve the system of these equations. The solution of the system represents the coordinates of the centroid.

Let AM be the median that joins the vertex A to the midpoint of BC, that is M.

To calculate the equation of the line AM, we'll use the formula:

(yM-yA)/(y-yA) = (xM-xA)/(x-xA)

We'll find the midpoint M:

xM = (xB+xC)/2

xM = (3+7)/2

xM = 5

yM = (yB+yC)/2

yM = (2+5)/2

yM = 7/2

(7/2-4)/(y-4) = (5-1)/(x-1)

(-1/4)/(y-4) = 4/(x-1)

4(y-4) = -x/4 + 1/4

y - 4 = -x/16 + 1/16

y = -x/16 + 1/16 + 4

y = -x/16 + 65/16 (1)

Let BN be the median that joins the vertex B to the midpoint of AC, that is N.

To calculate the equation of the line BN, we'll use the formula:

(yN-yB)/(y-yB) = (xN-xB)/(x-xB)

We'll find the midpoint N:

xN = (xA+xC)/2

xN = (1+7)/2

xN = 4

yN = (yA+yC)/2

yN = (4+5)/2

yN = 9/2

(9/2-2)/(y-2) = (4-3)/(x-3)

y - 2 = 5x/2 - 15/2

y = 5x/2 - 15/2 + 2

y = 5x/2 - 11/2 (2)

Now, we'll solve the system formed by equations of AM and BN.

We'll equate (1) and (2):

-x/16 + 65/16 = 5x/2 - 11/2

-x/8 + 65/8 = 5x - 11

We'll move the terms in x to the left:

-x/8 - 5x = -65/8 - 11

-41x/8 = -153/8

x = 153/41

y = 765/82 - 11/2

y = 314/82

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