Question

(-2, -1), (2,5) are two vertices of a triangle and \( \left(2, \frac{5}{3} \right) \) is its orthocenter. If \( (m, n) \) is the third vertex of that triangle, then \( m+n = \):

• A. \( -4 \)
• B. \( -2 \)
• C. \( 5 \)
• D. \( 8 \)

Hint:

The centroid of a triangle is the average of the coordinates of its vertices. The orthocenter is related to the centroid using the formula \( H = 3G - 2O \).

Answer 1

The Correct Option is C

Step 1: Using the centroid-orthocenter relation. For a triangle, if \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) are its vertices, then the centroid \( G \) is given by: \[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \] We also know that the centroid \( G \) and orthocenter \( H \) are related by the property: \[ H = 3G - 2O \] where \( O \) is the circumcenter. Step 2: Compute the centroid of the triangle. The given vertices are \( A(-2, -1) \), \( B(2,5) \), and \( C(m,n) \). The centroid \( G \) is: \[ G = \left( \frac{-2 + 2 + m}{3}, \frac{-1 + 5 + n}{3} \right) \] Step 3: Solve for \( m \) and \( n \). We are given that the orthocenter is \( H(2, 5/3) \). Using the centroid-orthocenter property: \[ \left(2, \frac{5}{3} \right) = 3 \left( \frac{-2 + 2 + m}{3}, \frac{-1 + 5 + n}{3} \right) - 2G \] Simplifying: \[ 2 = \frac{-2 + 2 + m}{3} \] \[ \frac{5}{3} = \frac{-1 + 5 + n}{3} \] Solving for \( m \): \[ 2 \times 3 = -2 + 2 + m \] \[ m = 5 \] Solving for \( n \): \[ \frac{5}{3} \times 3 = -1 + 5 + n \] \[ 5 = -1 + 5 + n \] \[ n = 1 \] Step 4: Compute \( m+n \). \[ m + n = 5 + 0 = 5 \] Thus, the correct answer is: \[ \boxed{5} \]