The coordinates of two vertices of a triangle are (6, 3) and (−1, 7) and its centroid is (1,5). Then the coordinate of third vertex is :

Question

Accepted Answer

To determine the coordinates of the third vertex of the triangle, we can use the formula for the centroid of a triangle. The centroid $ G(x, y) $ of a triangle with vertices at $ A(x_1, y_1) $, $ B(x_2, y_2) $, and $ C(x_3, y_3) $ is given by:

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

Given:
- The coordinates of two vertices are $ A(6, 3) $ and $ B(-1, 7) $.
- The centroid is $ G(1, 5) $.

We need to find the coordinates of the third vertex $ C(x_3, y_3) $.

Using the centroid formula, we have:

$$
1 = \frac{6 + (-1) + x_3}{3}
$$
$$
5 = \frac{3 + 7 + y_3}{3}
$$

Solving these equations:

1. For the x-coordinate:
   $$
   1 = \frac{5 + x_3}{3}
   $$
   $$
   3 = 5 + x_3
   $$
   $$
   x_3 = 3 - 5 = -2
   $$

2. For the y-coordinate:
   $$
   5 = \frac{10 + y_3}{3}
   $$
   $$
   15 = 10 + y_3
   $$
   $$
   y_3 = 15 - 10 = 5
   $$

Thus, the coordinates of the third vertex $ C $ are $ (-2, 5) $.

**Explanation for Option 3:**
- Option 3, which is $ (-2, 5) $, satisfies both the x and y conditions derived from the centroid formula. Therefore, it is the correct answer.

**Explanation for why other options are incorrect:**
1. **Option 1: (2, 5)**
   - The x-coordinate does not satisfy the equation $ 3 = 5 + x_3 $, as substituting $ x_3 = 2 $ gives $ 7 \neq 3 $.

2. **Option 2: (2, -5)**
   - Neither the x-coordinate nor the y-coordinate satisfies the equations derived from the centroid formula. Substituting $ x_3 = 2 $ gives $ 7 \neq 3 $, and substituting $ y_3 = -5 $ gives $ 5 \neq 15 $.

3. **Option 4: (-2, -5)**
   - The x-coordinate satisfies the equation, but the y-coordinate does not. Substituting $ y_3 = -5 $ gives $ 5 \neq 15 $.

Therefore, Option 3 is the only choice that correctly satisfies both conditions for the centroid, confirming it as the correct answer.