Math Problem Statement

The point p has coordinates (-12, -7). The point m has coordinates (-1, 4). The point m is the point on line L that is closest to p. Take q to be the reflection of p across L. What are the coordinates of q?

Solution

Let's solve this step by step:

  1. Understand the Problem:
    We are tasked to find the reflection of point p(12,7)p(-12, -7) across the line LL. The line LL passes through point m(1,4)m(-1, 4). The reflection is denoted as qq.

  2. Determine the Equation of Line LL:
    We need the slope of the line LL. Based on the given diagram and the coordinates mm, LL is perpendicular to the line segment joining pp and mm.

    The slope of the line segment joining p(12,7)p(-12, -7) and m(1,4)m(-1, 4) is: slope of pm=4(7)1(12)=1111=1\text{slope of pm} = \frac{4 - (-7)}{-1 - (-12)} = \frac{11}{11} = 1 Thus, the slope of line LL, being perpendicular to this, is: slope of L=1\text{slope of } L = -1

  3. Equation of Line LL:
    Using the point-slope form yy1=m(xx1)y - y_1 = m(x - x_1) and the point m(1,4)m(-1, 4), the equation of LL is: y4=1(x+1)y - 4 = -1(x + 1) Simplifying: y=x+3y = -x + 3

  4. Find the Reflection Point qq:
    To find the reflection of p(12,7)p(-12, -7), we:

    • Find the point on LL closest to pp (denoted as mm), already given in this case.
    • Use the reflection formula: q=(x,y) where x=2xmxp and y=2ymypq = (x', y') \text{ where } x' = 2x_m - x_p \text{ and } y' = 2y_m - y_p Here, m(1,4)m(-1, 4) and p(12,7)p(-12, -7).

    Substituting: x=2(1)(12)=2+12=10x' = 2(-1) - (-12) = -2 + 12 = 10 y=2(4)(7)=8+7=15y' = 2(4) - (-7) = 8 + 7 = 15

    Thus, q(10,15)q(10, 15).

Final Answer:

q=(10,15)q = (10, 15)

Let me know if you'd like further clarification or more detailed steps!


Relative Questions:

  1. How do we derive the slope of a line perpendicular to another?
  2. Can we calculate the distance between pp and qq?
  3. What is the general reflection formula in 2D geometry?
  4. How can we verify if a point lies on a given line equation?
  5. What happens to the reflection point if pp lies directly on the line?

Tip: Always confirm perpendicularity by multiplying the slopes; their product should be 1-1.

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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