The coordinates of the orthocenter for the triangle with vertices (-9, 9), (-6, 6), and (0, 6) are found to be (-6, 6) by determining perpendicular altitudes.
Given the vertices (-9, 9), (-6, 6), and (0, 6) of a triangle, we can find its orthocenter by determining the altitudes. The slope of the line segment connecting the points (-9, 9) and (-6, 6) is 1/3, and the slope of the line segment connecting (-6, 6) and (0, 6) is 0. The altitude from vertex A is perpendicular to side BC and has an undefined slope, as BC is a vertical line.
The altitude from vertex B is perpendicular to side AC and has a slope of -3. The equation of the altitude from B, using point-slope form with B(-6, 6), is y - 6 = -3(x + 6). Solving this system of equations, including the vertical line BC: x = -6, we find the coordinates of the orthocenter at the point of intersection.
By substituting x = -6 into the equation of the altitude from B, we get y = 6. Therefore, the orthocenter of the triangle is (-6, 6). This process illustrates how the slopes of the sides and perpendicular altitudes can be used to determine the orthocenter, a point where all three altitudes intersect in a triangle.
The probable question maybe:
What are the coordinates of the orthocenter for a triangle with vertices (-9, 9), (-6, 6), and (0, 6)?