Two vertices of a triangle are (5, -1) and (-2, 3). If orthocentre of the triangle is at (0, 0), then the third vertex of the triangle is :

Question

Accepted Answer

To determine why Option 4, (-4, -7), is the correct answer for the third vertex of the triangle, we need to consider the properties of the orthocenter in a triangle. The orthocenter is the point where the altitudes of a triangle intersect. Given that the orthocenter is at (0, 0), we can use this information to find the third vertex.

### Explanation for Option 4:

1. **Orthocenter Properties**: The orthocenter is the intersection of the altitudes of a triangle. For a triangle with vertices $ A(x_1, y_1) $, $ B(x_2, y_2) $, and $ C(x_3, y_3) $, the orthocenter $ H $ can be calculated using the slopes of the altitudes. Given that the orthocenter is at (0, 0), this point must satisfy the conditions derived from the altitudes of the triangle.

2. **Given Vertices**: The vertices given are $ A(5, -1) $ and $ B(-2, 3) $. We need to find the third vertex $ C(x_3, y_3) $.

3. **Using Slopes and Altitudes**: 
   - The slope of line AB is $\frac{3 - (-1)}{-2 - 5} = \frac{4}{-7} = -\frac{4}{7}$.
   - The altitude from C to AB must be perpendicular to AB, so its slope is the negative reciprocal, $\frac{7}{4}$.

4. **Equation of Altitude from C**: Since the orthocenter is at (0, 0), the line from C to the orthocenter must have a slope of $\frac{7}{4}$. Therefore, the equation of the line is $ y = \frac{7}{4}x $.

5. **Verification of Option 4**:
   - If $ C = (-4, -7) $, substituting into the line equation $ y = \frac{7}{4}x $ gives:
     $$
     -7 = \frac{7}{4}(-4)
     $$
     $$
     -7 = -7
     $$
   - This satisfies the equation, confirming that the line from C to the orthocenter is indeed perpendicular to AB.

### Explanation for Incorrect Options:

1. **Option 1 (7, 4)**:
   - Substituting into the line equation $ y = \frac{7}{4}x $ gives:
     $$
     4 \neq \frac{7}{4}(7)
     $$
     $$
     4 \neq \frac{49}{4}
     $$
   - This does not satisfy the equation, so it cannot be the correct vertex.

2. **Option 2 (-4, 7)**:
   - Substituting into the line equation $ y = \frac{7}{4}x $ gives:
     $$
     7 \neq \frac{7}{4}(-4)
     $$
     $$
     7 \neq -7
     $$
   - This does not satisfy the equation, so it cannot be the correct vertex.

3. **Option 3 (4, -7)**:
   - Substituting into the line equation $ y = \frac{7}{4}x $ gives:
     $$
     -7 \neq \frac{7}{4}(4)
     $$
     $$
     -7 \neq 7
     $$
   - This does not satisfy the equation, so it cannot be the correct vertex.

In conclusion, Option 4 (-4, -7) is the only choice that satisfies the conditions for the orthocenter and the properties of the triangle, making it the correct answer.