Math Problem Statement
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Solution
To solve the problem of finding the orthocenter, we first need to identify the vertices of the triangle from the coordinate plane in the image.
Here’s how to proceed:
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Identify the vertices: From the grid, you can see that the triangle's vertices are at:
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Understand the orthocenter: The orthocenter of a triangle is the point where all three altitudes intersect. An altitude is a line segment from a vertex perpendicular to the opposite side.
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Altitudes of the triangle:
- The altitude from vertex is along the vertical side of the triangle (the y-axis), which means it is already on the y-axis.
- The altitude from vertex would be perpendicular to the line between and , which is vertical, so this altitude lies on the x-axis.
- The third altitude is drawn from vertex and is perpendicular to the base of the triangle (the line from to ).
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Finding the orthocenter: In this specific triangle, which is a right triangle, the orthocenter lies at the right angle vertex. Since the right angle is at vertex , the orthocenter is at this point.
So, the coordinates of the orthocenter are .
Would you like more details on this? Here are five follow-up questions for further exploration:
- What is the difference between the orthocenter and the centroid of a triangle?
- How do you find the orthocenter for an obtuse triangle?
- What is the relationship between the orthocenter and other triangle centers (like the circumcenter or incenter)?
- How do you find the equation of an altitude for non-right triangles?
- Why is the orthocenter the same as the vertex for a right triangle?
Tip: The orthocenter is always inside the triangle for an acute triangle but lies outside for obtuse triangles.
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Math Problem Analysis
Mathematical Concepts
Geometry
Coordinate Geometry
Triangle Centers
Orthocenter
Formulas
Altitude of a Triangle
Perpendicular Lines on a Coordinate Plane
Theorems
Orthocenter Theorem for Right Triangles
Suitable Grade Level
Grades 9-12
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