Triangle \( A B C \), with the following coordinate plane. - \( A B \) is on a vertical line. - P is a right angle. - The slope of AC is 5 . What are possible coordinates for

To find possible coordinates for triangle \( ABC \) given the conditions, let's define the points based on the information provided: 1. **Point A**: Let's place point \( A \) at coordinates \( (x_A, y_A) \). 2. **Point B**: Since \( AB \) is on a vertical line, point \( B \) will have the same \( x \)-coordinate as point \( A \). Therefore, we can denote point \( B \) as \( (x_A, y_B) \). 3. **Point C**: We need to find the coordinates of point \( C \) such that angle \( P \) (which we assume is at point \( A \)) is a right angle, and the slope of line \( AC \) is 5. ### Step 1: Determine the slope of line \( AC \) The slope of line \( AC \) is given by the formula: \[ \text{slope of } AC = \frac{y_C - y_A}{x_C - x_A} \] Given that the slope is 5, we can set up the equation: \[ \frac{y_C - y_A}{x_C - x_A} = 5 \] This can be rearranged to: \[ y_C - y_A = 5(x_C - x_A) \] Thus, we can express \( y_C \) in terms of \( x_C \): \[ y_C = 5(x_C - x_A) + y_A \] ### Step 2: Determine the coordinates of point B Since \( AB \) is vertical, we can choose any \( y_B \) value. For simplicity, let's assume \( y_B = y_A + d \) where \( d \) is some positive distance. Thus, point \( B \) can be represented as: \[ B = (x_A, y_A + d) \] ### Step 3: Determine the coordinates of point C To ensure that angle \( P \) at point \( A \) is a right angle, we need to ensure that the lines \( AB \) and \( AC \) are perpendicular. The slope of line \( AB \) is undefined (vertical), and for \( AC \) to be perpendicular to \( AB \), the slope of \( AC \) must be 0 (horizontal). However, since we have a slope of 5, we need to ensure that the coordinates of \( C \) are such that the angle at \( A \) remains a right angle. ### Example Coordinates Let's choose specific coordinates for \( A \): - Let \( A = (0, 0) \). - Let \( B = (0, 3) \) (choosing \( d = 3 \)). - Now, we can find \( C \) using the slope condition. If we let \( x_C = 1 \): \[ y_C = 5(1 - 0) + 0 = 5 \] Thus, we have: - \( A = (0, 0) \) - \( B = (0, 3) \) - \( C = (1, 5) \) ### Conclusion The possible coordinates for triangle \( ABC \) could be: - \( A(0, 0) \) - \( B(0, 3) \) - \( C(1, 5) \) This satisfies the conditions of the problem: \( AB \) is vertical, angle \( P \) is a right angle, and the slope of \( AC \) is 5.