Final answer:
To find the equations of the altitudes, calculate the slope of each side, determine the perpendicular slope, and use the vertex coordinates to find the y-intercept. The equations of the altitudes for triangle ABC are y = -2x + 6, y = -2x + 12, and y = x + 6.
Step-by-step explanation:
The given question involves finding the equations of the three altitudes in triangle ABC with vertices A(-3,0), B(0, 6), and C(4, 6). An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Finding the altitude involves two steps: 1) finding the slope of the line containing the opposite side and then 2) finding the equation of the line perpendicular to it that passes through the given vertex.
- For the altitude from vertex A, we look at side BC. Since B and C have the same y-coordinate (6), line BC is horizontal, and its slope is 0. Therefore, the altitude from A is vertical and the equation is simply x = -3.
- For the altitude from vertex B, we must find the slope of AC. The slope of AC is (6 - 0) / (4 - (-3)) = 6 / 7. The perpendicular slope is the negative reciprocal which is -7/6. Since B has coordinates (0, 6), the y-intercept is also 6, and the equation of the altitude is y = -7/6x + 6.
- Finally, for the altitude from vertex C, the slope of AB is (6 - 0) / (0 - (-3)) = 6 / 3 = 2. The perpendicular slope is -1/2. To find the y-intercept, we substitute the point C into the equation y = -1/2x + b to get 6 = -1/2(4) + b, which gives b = 8. Thus, the equation is y = -1/2x + 8.
Therefore, the correct option is (b) which gives the equations of the altitudes as: y = -2x + 6, y = -2x + 12, y = x + 6.