Math Problem Statement

Graph triangle RST with vertices R(4, 1), S(7, 3), and T(6, 4) and its image after the glide reflection. Also, determine the number of lines of symmetry for given figures.

Solution

Let's solve the questions based on the provided image.

Question 17: Graph and Glide Reflection

Given:

  • Vertices of ΔRST: R(4,1),S(7,3),T(6,4)R(4, 1), S(7, 3), T(6, 4)
  • Translation: (x,y)(x,y1)(x, y) \to (x, y - 1)
  • Reflection: Reflection across the yy-axis.
  1. Translation: Translate each vertex of ΔRST\Delta RST by reducing the yy-coordinate by 1:

    • R(4,1)R(4,0)R(4, 1) \to R'(4, 0)
    • S(7,3)S(7,2)S(7, 3) \to S'(7, 2)
    • T(6,4)T(6,3)T(6, 4) \to T'(6, 3)
  2. Reflection in the yy-axis: Reflect each R,S,TR', S', T' over the yy-axis:

    • R(4,0)R(4,0)R'(4, 0) \to R''(-4, 0)
    • S(7,2)S(7,2)S'(7, 2) \to S''(-7, 2)
    • T(6,3)T(6,3)T'(6, 3) \to T''(-6, 3)

Final vertices of the reflected triangle are:

  • R(4,0)R''(-4, 0), S(7,2)S''(-7, 2), T(6,3)T''(-6, 3).

Question 21: Number of Lines of Symmetry for the Trapezoid

  • The trapezoid shown is isosceles (sides are symmetric about the vertical axis).
  • Number of Lines of Symmetry: 1 (the vertical line passing through the midpoint of the top and bottom sides).

Question 22: Number of Lines of Symmetry for the Cross Shape

  • The cross shape is symmetric both vertically and horizontally and along both diagonals.
  • Number of Lines of Symmetry: 4.

Would you like further details, graphs, or visual explanations? Let me know!


Here are 5 related questions to expand your understanding:

  1. How do you graph transformations like reflections and translations on a coordinate plane?
  2. Can every shape have a line of symmetry? Why or why not?
  3. How do diagonal lines of symmetry differ from vertical or horizontal ones?
  4. What are the effects of combining transformations, such as translation followed by reflection?
  5. How do you verify if a figure has a specific number of lines of symmetry?

Tip: Always perform transformations step-by-step to avoid errors in composite operations!

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Math Problem Analysis

Mathematical Concepts

Transformations
Glide Reflection
Symmetry

Formulas

Translation: (x, y) → (x, y - 1)
Reflection across y-axis: (x, y) → (-x, y)

Theorems

Reflection Theorem
Properties of Symmetry

Suitable Grade Level

Grades 8-10