Question

If \( (a, \beta) \) is the orthocenter of the triangle with the vertices \( A(2, 5), B(1, 5), C(1, 4) \), then \( a + \beta = \):

• A. 6
• B. 5
• C. \( \frac{7}{2} \)
• D. \( \frac{5}{2} \)

Hint:

To find the orthocenter of a triangle, calculate the altitudes from two vertices and find their intersection point. The orthocenter will be the point where the altitudes meet.

Answer 1

The Correct Option is A

We are given that \( (a, \beta) \) is the orthocenter of the triangle with vertices \( A(2, 5), B(1, 5), C(1, 4) \). To find \( a + \beta \), we need to first calculate the equation of the altitudes of the triangle and find the orthocenter.
Step 1: To find the orthocenter, we first find the slopes of the sides of the triangle.
- The slope of side \( AB \) is: \[ m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{5 - 5}{1 - 2} = 0 \] Since the slope of \( AB \) is 0, the altitude from point \( C \) is a vertical line, and its equation is \( x = 1 \). 
- The slope of side \( BC \) is: \[ m_{BC} = \frac{y_C - y_B}{x_C - x_B} = \frac{4 - 5}{1 - 1} = \infty \] Since the slope of \( BC \) is undefined, the altitude from point \( A \) is a horizontal line, and its equation is \( y = 5 \).
Step 2: Now, the orthocenter is the point of intersection of these two altitudes, which are given by the equations: \[ x = 1 \quad {and} \quad y = 5 \] Thus, the coordinates of the orthocenter are \( (1, 5) \), so \( a = 1 \) and \( \beta = 5 \). 
Step 3: Finally, we calculate \( a + \beta \): \[ a + \beta = 1 + 5 = 6 \] Thus, the value of \( a + \beta \) is 6.