Math Problem Statement
Solution
Let's solve this step by step:
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Understand the Problem:
We are tasked to find the reflection of point across the line . The line passes through point . The reflection is denoted as . -
Determine the Equation of Line :
We need the slope of the line . Based on the given diagram and the coordinates , is perpendicular to the line segment joining and .The slope of the line segment joining and is: Thus, the slope of line , being perpendicular to this, is:
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Equation of Line :
Using the point-slope form and the point , the equation of is: Simplifying: -
Find the Reflection Point :
To find the reflection of , we:- Find the point on closest to (denoted as ), already given in this case.
- Use the reflection formula: Here, and .
Substituting:
Thus, .
Final Answer:
Let me know if you'd like further clarification or more detailed steps!
Relative Questions:
- How do we derive the slope of a line perpendicular to another?
- Can we calculate the distance between and ?
- What is the general reflection formula in 2D geometry?
- How can we verify if a point lies on a given line equation?
- What happens to the reflection point if lies directly on the line?
Tip: Always confirm perpendicularity by multiplying the slopes; their product should be .
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Reflections
Slopes of Lines
Point-Line Distance
Formulas
Slope formula: (y2 - y1) / (x2 - x1)
Point-slope form: y - y1 = m(x - x1)
Reflection formula: q = (2xm - xp, 2ym - yp)
Theorems
Reflection Theorem in Coordinate Geometry
Properties of Perpendicular Lines
Suitable Grade Level
Grades 10-12