Final answer:
To find the coordinates of the orthocentre of a triangle, we first need to find the equations of the altitudes and then determine their point of intersection. The altitudes of the given triangle are x = 1, y = -3x + 9, and y = -x + 7. When we substitute x = 1 into the equations of the other two altitudes, we find that y = 6 and y = 6, respectively. Therefore, the orthocentre of the triangle is (1,6).
Step-by-step explanation:
To find the coordinates of the orthocentre of a triangle, we first need to find the equations of the altitudes. The altitude of a triangle is a line segment from a vertex of the triangle that is perpendicular to the opposite side. Once we have the equations of the altitudes, we can find their point of intersection, which will be the orthocentre.
Let's start by finding the equations of the altitudes.
The given triangle has vertices (1,2), (2,3), and (4,3).
The altitude from the vertex (1,2) is perpendicular to the line passing through the points (2,3) and (4,3). The slope of this line is (3-3)/(2-4) = 0. The slope of the altitude will be the negative reciprocal of this slope, which is undefined since the line is vertical. Therefore, the equation of the altitude from (1,2) is x = 1.
The altitude from the vertex (2,3) is perpendicular to the line passing through the points (1,2) and (4,3). The slope of this line is (3-2)/(4-1) = 1/3. The slope of the altitude will be the negative reciprocal of this slope, which is -3. We can use the point-slope form of a linear equation to find the equation of the altitude. Let's use the point (2,3) as the starting point. The equation of the altitude from (2,3) is y - 3 = -3(x - 2), which simplifies to y = -3x + 9.
The altitude from the vertex (4,3) is perpendicular to the line passing through the points (1,2) and (2,3). The slope of this line is (3-2)/(2-1) = 1. The slope of the altitude will be the negative reciprocal of this slope, which is -1. We can use the point-slope form of a linear equation to find the equation of the altitude. Let's use the point (4,3) as the starting point. The equation of the altitude from (4,3) is y - 3 = -1(x - 4), which simplifies to y = -x + 7.
Now that we have the equations of the altitudes, we can find their point of intersection, which will be the orthocentre. Since the equation of the altitude from (1,2) is x = 1, it intersects the other two altitudes at x = 1. Substituting this value into the equations of the other two altitudes, we find that y = 6 and y = 6, respectively. Therefore, the orthocentre of the triangle is (1,6).