# A triangle has corners at (9 ,1 ), (6 ,7 ), and (3 ,2 ). How far is the triangle's centroid from the origin?

Apr 23, 2018

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The triangle's Centroid is color(red)("6.3 Units" away from the Origin.

#### Explanation:

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color(green)("Step 1"

Construct a triangle ABC with Vertices as given:

color(green)("Step 2"

A Centroid of a triangle ABC is the point where the three medians of the triangle meet.

A median of a triangle is a line segment from one vertex to the mid point on the opposite side of the triangle.

Mark the Mid-Points of the line segments $A B , B C \mathmr{and} A C$.

Call these mid-points $D , E \mathmr{and} F$.

Mid-Point of Side $\overline{A B}$ is color(blue)(D.

Mid-Point of Side $\overline{B C}$ is color(blue)(E.

Mid-Point of Side $\overline{A C}$ is color(blue)(F.

Join Side $\overline{A B}$ to the color(red)("Vertex C".

Join Side $\overline{B C}$ to the color(red)("Vertex A".

Join Side $\overline{A C}$ to the color(red)("Vertex B".

You can see that the all the three Medians intersect at a point.

This point is called the Centroid.

color(green)("Step 3"

Identify the Origin (O) with coordinates $\left(0 , 0\right)$.

Mark the Centroid and name it as Centroid.

Connect the Origin and the Centroid.

Measure the length of the line segment joining the Origin and the Centroid.

The magnitude $\approx \text{ 6.227 Units}$.

You can write your final solution as:

The triangle's Centroid is color(red)("6.3 Units" away from the Origin.

An interesting observation to note:

The three line segments $\overline{A B} , \overline{B C} \mathmr{and} \overline{A C}$ intersect at one point if and only if:

color(blue)(bar(AD)/bar(BD) = 1

color(blue)(bar(BE)/bar(CE) = 1

color(blue)(bar(CF)/bar(AF) = 1

This property is known as Ceva's Theorem.

Hope it helps.