5-Coordinate Geometry and the Centroid - Kuta Software - Free Printable

To solve this problem, we need to find the coordinates of the centroid of each triangle shown on the coordinate plane.

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🔷 What is a Centroid?

The centroid of a triangle is the point where the three medians intersect. It is also the average of the coordinates of the triangle’s three vertices.

If a triangle has vertices at:
- $ A(x_1, y_1) $
- $ B(x_2, y_2) $
- $ C(x_3, y_3) $

Then the centroid $ G $ has coordinates:

$$
G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
$$

We'll apply this formula to each triangle.

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Let’s go through each triangle one by one.

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1) First Triangle (Top-left)

Vertices:
- $ A(-3, 0) $
- $ B(0, 0) $
- $ C(0, 4) $

Centroid:
$$
x = \frac{-3 + 0 + 0}{3} = \frac{-3}{3} = -1 \\
y = \frac{0 + 0 + 4}{3} = \frac{4}{3}
$$

✔ Centroid: $ (-1, \frac{4}{3}) $

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2) Second Triangle (Top-right)

Vertices:
- $ A(-5, 0) $
- $ B(1, 0) $
- $ C(0, 6) $

Centroid:
$$
x = \frac{-5 + 1 + 0}{3} = \frac{-4}{3} \\
y = \frac{0 + 0 + 6}{3} = \frac{6}{3} = 2
$$

✔ Centroid: $ \left( -\frac{4}{3}, 2 \right) $

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3) Third Triangle (Middle-left)

Vertices:
- $ M(-4, 4) $
- $ P(4, 4) $
- $ Z(0, -2) $

Centroid:
$$
x = \frac{-4 + 4 + 0}{3} = \frac{0}{3} = 0 \\
y = \frac{4 + 4 + (-2)}{3} = \frac{6}{3} = 2
$$

✔ Centroid: $ (0, 2) $

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4) Fourth Triangle (Middle-right)

Vertices:
- $ A(-3, 0) $
- $ B(3, 0) $
- $ C(0, -4) $

Centroid:
$$
x = \frac{-3 + 3 + 0}{3} = \frac{0}{3} = 0 \\
y = \frac{0 + 0 + (-4)}{3} = \frac{-4}{3}
$$

✔ Centroid: $ \left( 0, -\frac{4}{3} \right) $

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5) Fifth Triangle (Bottom-left)

Vertices:
- $ S(-2, 4) $
- $ R(0, 0) $
- $ T(6, 0) $

Centroid:
$$
x = \frac{-2 + 0 + 6}{3} = \frac{4}{3} \\
y = \frac{4 + 0 + 0}{3} = \frac{4}{3}
$$

✔ Centroid: $ \left( \frac{4}{3}, \frac{4}{3} \right) $

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6) Sixth Triangle (Bottom-right)

Vertices:
- $ M(-4, 4) $
- $ K(4, 4) $
- $ L(0, -2) $

Centroid:
$$
x = \frac{-4 + 4 + 0}{3} = \frac{0}{3} = 0 \\
y = \frac{4 + 4 + (-2)}{3} = \frac{6}{3} = 2
$$

✔ Centroid: $ (0, 2) $

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✔ Final Answers:

| Problem | Centroid Coordinates |
|--------|------------------------|
| 1) | $ (-1, \frac{4}{3}) $ |
| 2) | $ \left( -\frac{4}{3}, 2 \right) $ |
| 3) | $ (0, 2) $ |
| 4) | $ \left( 0, -\frac{4}{3} \right) $ |
| 5) | $ \left( \frac{4}{3}, \frac{4}{3} \right) $ |
| 6) | $ (0, 2) $ |

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Let me know if you'd like these written as decimals or simplified further!