Math Problem Statement
Solve part 2
Solution
Let's break down and solve part (ii) of the problem.
We are given:
Problem:
The point lies on line such that line is perpendicular to line . We are asked to calculate the x-coordinate of .
Steps:
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Find the slope of : The slope between two points and is given by: Substituting the coordinates of and :
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Find the slope of : Since is perpendicular to , the slopes of the two lines satisfy: Hence, the slope of , , is:
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Equation of line : The equation of a line passing through two points can be written as: Using the slope and point : Simplifying: So, the equation of line is:
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Equation of line : The line passes through and has slope , so its equation is: Simplifying: So, the equation of line is:
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Intersection of lines and : To find the coordinates of , we need to solve the system of equations of lines and .
Set the equations equal to each other: Let's multiply through by 42 to eliminate the denominators: Simplifying both sides: Combine like terms: Solving for :
Thus, the x-coordinate of point is approximately .
Would you like further clarification or additional details on any steps? Here are some related questions:
- How do we verify if a triangle is isosceles using the distance formula?
- Can you explain the process for finding the y-coordinate of ?
- How do perpendicular lines relate in terms of slopes?
- What are alternative methods for finding the intersection of two lines?
- How can we determine the area of triangle using its vertices?
Tip: When solving for the intersection of two lines, always check for computational accuracy by verifying the solution in the original equations.
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Math Problem Analysis
Mathematical Concepts
Analytic Geometry
Perpendicular Slopes
Intersection of Lines
Formulas
Slope of a line: m = (y2 - y1) / (x2 - x1)
Perpendicular slopes: m1 * m2 = -1
Equation of a line: y - y1 = m(x - x1)
Theorems
Perpendicular Line Theorem
Slope-Intercept Theorem
Suitable Grade Level
Grades 9-12
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