Math Problem Statement
if vertices of a triangle ABC are A(1,2) B( 3,4) and C(-5,3) and its orthocentre is O, then the reflection of orthocentre of triangle OAB with respect to line y=x will be
Solution
To find the reflection of the orthocenter of triangle with respect to the line , we'll first need to determine the orthocenter of triangle . Then, we reflect this orthocenter across the line .
Step 1: Find the orthocenter of triangle .
Given the vertices:
The slopes of the sides opposite to each vertex are:
- Slope of :
- Slope of :
- Slope of :
Now, we find the equations of the altitudes:
- The slope of the altitude from is perpendicular to , so it will have a slope of . Its equation using point-slope form is:
- The slope of the altitude from is perpendicular to , so it will have a slope of . Its equation using point-slope form is:
To find the orthocenter , solve the system of equations: Substituting into one of the altitude equations: Thus, the orthocenter is .
Step 2: Reflect the orthocenter across the line .
To reflect a point across the line , swap the coordinates. The reflection of is:
Final Answer:
The reflection of the orthocenter of triangle with respect to the line is .
Would you like further details or have any questions about the steps? Here are five related questions to explore further:
- How do you calculate the centroid of a triangle given its vertices?
- What is the significance of the circumcenter in a triangle?
- How do you find the equation of a line given a point and a slope?
- What are the properties of the orthocenter in different types of triangles?
- How do you reflect a point across a line in general?
Tip: When solving for the intersection of two lines, using substitution or elimination methods is very efficient.
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Math Problem Analysis
Mathematical Concepts
Geometry
Coordinate Geometry
Reflections
Formulas
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Theorems
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Suitable Grade Level
High School
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