ABC is a triangle and G (4, 3) is the centroid of the triangle. If A = (1, 3), B = (4, b) and C = (a, 1), find 'a' and 'b'.
Find the length of side BC.

Answer

To solve the problem step by step, we will first find the values of 'a' and 'b' using the properties of the centroid of a triangle. Then, we will calculate the length of side BC.

### Step 1: Use the centroid formula to find 'a' and 'b'.

The coordinates of the centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) are given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
Given that \( G(4, 3) \), \( A(1, 3) \), \( B(4, b) \), and \( C(a, 1) \), we can set up the following equations:

1. For the x-coordinates:
\[ 4 = \frac{1 + 4 + a}{3} \]

2. For the y-coordinates:
\[ 3 = \frac{3 + b + 1}{3} \]

### Step 2: Solve for 'a'.

From the x-coordinate equation:
\[ 4 = \frac{1 + 4 + a}{3} \]
Multiply both sides by 3:
\[ 12 = 1 + 4 + a \]
Combine the constants:
\[ 12 = 5 + a \]
Now, isolate 'a':
\[ a = 12 - 5 = 7 \]

### Step 3: Solve for 'b'.

From the y-coordinate equation:
\[ 3 = \frac{3 + b + 1}{3} \]
Multiply both sides by 3:
\[ 9 = 3 + b + 1 \]
Combine the constants:
\[ 9 = 4 + b \]
Now, isolate 'b':
\[ b = 9 - 4 = 5 \]

### Step 4: Find the length of side BC.

Now that we have \( B(4, 5) \) and \( C(7, 1) \), we can find the length of side BC using the distance formula:
\[ BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of B and C:
\[ BC = \sqrt{(4 - 7)^2 + (5 - 1)^2} \]
Calculating the differences:
\[ BC = \sqrt{(-3)^2 + (4)^2} \]
Calculating the squares:
\[ BC = \sqrt{9 + 16} \]
Adding the results:
\[ BC = \sqrt{25} \]
Finally, taking the square root:
\[ BC = 5 \]

### Final Answers:
- The values of \( a \) and \( b \) are \( a = 7 \) and \( b = 5 \).
- The length of side \( BC \) is \( 5 \) units.