Math Problem Statement

Determine the equation of the line of reflection given the slope of the line (-1) and points of reflected triangles.

Solution

From the provided image, the question involves identifying the equation of the line of reflection. Here is how to determine it step-by-step:

  1. Given Information:

    • The slope of the line of reflection is 1-1.
    • The dotted lines connect corresponding points from the reflected triangles.
    • The reflection line passes through the midpoints of these segments.
  2. Identify the midpoint of one pair of corresponding points:

    • From the graph, the coordinates of one point are approximately (3,3)(-3, 3) and its reflection is at (1,1)(1, -1).
    • The midpoint is calculated as: Midpoint=(x1+x22,y1+y22)=(3+12,3+(1)2)=(1,1).\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{-3 + 1}{2}, \frac{3 + (-1)}{2}\right) = (-1, 1).
  3. Equation of the line with slope 1-1 passing through (1,1)(-1, 1): Using the point-slope form of a line: yy1=m(xx1),y - y_1 = m(x - x_1), where m=1m = -1 (slope) and (1,1)(-1, 1) is the point: y1=1(x+1).y - 1 = -1(x + 1). Simplify: y=x.y = -x.

  4. Final Answer: The equation of the line of reflection is: y=x.y = -x.

Would you like further clarification or details about this process? Here are related questions to deepen understanding:

  1. How do you verify a line is the correct reflection line for two given points?
  2. Can the method used here be generalized for reflections over any line?
  3. Why is the midpoint of corresponding points always on the line of reflection?
  4. How does the slope of the line of reflection relate to the slopes of the connecting dotted lines?
  5. How do transformations like reflection relate to congruence in geometry?

Tip: To confirm the equation of the line of reflection, check that it equidistantly bisects the segments connecting all pairs of reflected points.

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Math Problem Analysis

Mathematical Concepts

Reflections in Geometry
Lines and Slopes
Point-Slope Form of a Line

Formulas

Midpoint formula: Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Point-slope form: y - y1 = m(x - x1)

Theorems

Reflection theorem: The line of reflection bisects the line segment connecting corresponding points of the reflected figures perpendicularly.

Suitable Grade Level

Grades 8-10