(0,5) (3,1) (8,1) find the orthocenter
To find the orthocenter of a triangle, we need to use the coordinates of the vertices of the triangle. The orthocenter is the point where the altitudes of the triangle intersect.
Given the vertices of the triangle as (0, 5), (3, 1), and (8, 1), let's proceed step-by-step to find the orthocenter.
Step 1: Find the slopes of two sides of the triangle:
- The slope of the line joining the points (0, 5) and (3, 1) is given by (y2 - y1) / (x2 - x1) = (1 - 5) / (3 - 0) = -4/3.
- The slope of the line joining the points (3, 1) and (8, 1) is given by (y2 - y1) / (x2 - x1) = (1 - 1) / (8 - 3) = 0.
Step 2: Find the slopes of the corresponding altitudes:
- Perpendicular lines have negative reciprocal slopes. Therefore, the slope of an altitude of the side joining (0, 5) and (3, 1) would be 3/4 (negative reciprocal of -4/3).
- The slope of an altitude of the side joining (3, 1) and (8, 1) would be undefined (since the slope of the side is 0, meaning it is a horizontal line).
Step 3: Find the equations of the altitudes:
- We have one altitude perpendicular to the side joining (0, 5) and (3, 1). To find its equation, we can use the point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Using the point (3, 1) and slope 3/4, the equation of the altitude is y - 1 = 3/4(x - 3), which simplifies to y = 3/4x - 9/4.
Step 4: Find the intersection point of the two altitudes:
- Since one altitude is horizontal (slope undefined), it is a line parallel to the x-axis, which means its equation is y = constant value. In this case, the line is y = 1.
- We need to find the x-coordinate where the equation of the other altitude, y = 3/4x - 9/4, intersects with y = 1. Setting the two equations equal to each other, we have 3/4x - 9/4 = 1.
Solving this equation, we find x = 13/3.
Step 5: Substitute the x-coordinate into one of the altitude equations to find the y-coordinate:
- Substituting x = 13/3 into y = 3/4x - 9/4, we have y = 3/4 * (13/3) - 9/4, which simplifies to y = -5/4.
Therefore, the coordinates of the orthocenter of the triangle with vertices (0, 5), (3, 1), and (8, 1) are (13/3, -5/4).
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