The centroid of $\triangle ABC$ is $G$. Find $BG.$

[asy]
unitsize (1 cm);

pair A, B, C, G;

A = (5,0);
B = (0,0);
C = (0,3);
G = (A + B + C)/3;

draw(A--B--C--cycle);
draw(A--G);
draw(B--G);
draw(C--G);
draw(rightanglemark(A,B,C,8));

label("$A$", A, SE);
label("$B$", B, SW);
label("$C$", C, NW);
label("$G$", G, NE);
label("$3$", (C + G)/2, SW, red);
label("$4$", interp(A,G,0.7), SW, red);
[/asy]


We are tasked with finding $ BG $, the distance between point $ B $ and the centroid $ G $ of $ \triangle ABC $.

### Step 1: Coordinates of Points
The given coordinates of the vertices of the triangle are:
- $ A(5, 0) $
- $ B(0, 0) $
- $ C(0, 3) $

### Step 2: Centroid Formula
The centroid $ G $ of a triangle with vertices $ A(x_1, y_1) $, $ B(x_2, y_2) $, and $ C(x_3, y_3) $ is given by the formula:
\[
G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\]
Substituting the coordinates of $ A $, $ B $, and $ C $:
\[
G = \left( \frac{5 + 0 + 0}{3}, \frac{0 + 0 + 3}{3} \right) = \left( \frac{5}{3}, 1 \right)
\]

### Step 3: Distance Formula
The distance between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Using the coordinates $ B(0, 0) $ and $ G\left(\frac{5}{3}, 1\right) $, we can calculate $ BG $:
\[
BG = \sqrt{\left( \frac{5}{3} - 0 \right)^2 + \left( 1 - 0 \right)^2} = \sqrt{\left( \frac{5}{3} \right)^2 + 1^2}
\]
\[
BG = \sqrt{\frac{25}{9} + 1} = \sqrt{\frac{25}{9} + \frac{9}{9}} = \sqrt{\frac{34}{9}} = \frac{\sqrt{34}}{3}
\]

Thus, the distance $ BG $ is $ \frac{\sqrt{34}}{3} $.

Would you like further details or have any questions?

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Here are 5 related questions:
1. What is the formula for finding the centroid of a triangle?
2. How would you calculate the centroid if the triangle had different coordinates?
3. How does the centroid divide each median in a triangle?
4. Can you explain why the centroid lies inside the triangle for all triangles?
5. What is the significance of the centroid in applications like physics (center of mass)?

**Tip:** The centroid divides each median of a triangle in the ratio 2:1, with the longer segment being closer to the vertex.

The centroid of $\triangle ABC$ is $G$. Find $BG.$

Learn how to find the distance BG from vertex B to the centroid G of triangle ABC using the centroid and distance formulas. This problem covers key geometry concepts suitable for students in Grades 9-11.

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