If centroid of the points (3, 5), (6, y) and (-3, 4) is (x, y), then find the values of x and y.

Question

Accepted Answer

To find the values of x and y, we use the formula for the centroid of three points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, which is given by:
Centroid $(X, Y) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}ight)$

Given points are $(3, 5)$, $(6, y)$, and $(-3, 4)$.
The centroid is given as $(x, y)$.

**Step 1: Calculate the x-coordinate of the centroid.**
Using the formula for the x-coordinate:
$x = \frac{3 + 6 + (-3)}{3}$
$x = \frac{3 + 6 - 3}{3}$
$x = \frac{9 - 3}{3}$
$x = \frac{6}{3}$
$x = 2$

**Step 2: Calculate the y-coordinate of the centroid.**
Using the formula for the y-coordinate:
$y = \frac{5 + y + 4}{3}$
Now, we need to solve this equation for $y$:
$3y = 5 + y + 4$
$3y = 9 + y$
Subtract $y$ from both sides:
$3y - y = 9$
$2y = 9$
Divide by 2:
$y = \frac{9}{2}$

**Step 3: Combine the calculated values.**
From the calculations, we found $x=2$ and $y=\frac{9}{2}$.

**Justification for Option 3:**
Option 3 states $x=2, y=\frac{9}{2}$. This perfectly matches our calculated values for x and y. Therefore, Option 3 is the correct answer.

**Explanation for why other options are incorrect:**

*   **Option 1: $x=\frac{9}{2}, y=3$**
    This option is incorrect because our calculated value for $x$ is $2$, not $\frac{9}{2}$, and our calculated value for $y$ is $\frac{9}{2}$, not $3$.

*   **Option 2: $x=2, y=3$**
    This option is incorrect because while the x-coordinate ($x=2$) matches our calculation, the y-coordinate ($y=3$) does not. Our calculated y-coordinate is $\frac{9}{2}$.

*   **Option 4: $x=3, y=2$**
    This option is incorrect because our calculated value for $x$ is $2$, not $3$, and our calculated value for $y$ is $\frac{9}{2}$, not $2$.

Based on the detailed calculation, the values of x and y are $2$ and $\frac{9}{2}$ respectively, which corresponds to Option 3.

The final answer is $\boxed{	ext{Option 3}}$