Math Problem Statement

14. Parallelogram KLMN is plotted on the grid. (a) Draw the lines x = -1 and y = -2. (b) Draw the image of KLMN after a reflection in the line x = -1. Label the image K'L'M'N'. (c) Draw the image of K'L'M'N' after a reflection in the line y = -2. Label its image K''L''M''N''. 15. Triangle ABC has vertices at A(-8, -6), B(-3, -6), and C(-2, -3). (a) Draw the image of ΔABC after a reflection over the y-axis. Label its image ΔA'B'C'. List the coordinates of its vertices below. (b) Draw the image of ΔA'B'C' after a rotation of 180° around the origin. Label its image ΔA''B''C''. List the coordinates of its vertices below. (c) Draw the image of ΔA''B''C'' after a translation under the rule (x, y) → (x, y + 4). Label its image ΔA'''B'''C'''. List the coordinates of its vertices below. (d) What single transformation would map ΔABC to ΔA'''B'''C'''?

Solution

Here are the solutions to the questions provided:


Question 14 (Parallelogram KLMN):

Part (a):

  • Draw the lines x=1x = -1 (a vertical line) and y=2y = -2 (a horizontal line) on the grid.
    • These lines are used as axes of reflection in parts (b) and (c).

Part (b):

Reflection of KLMN over x=1x = -1:

  • When reflecting a point over a vertical line x=kx = k, the xx-coordinate changes symmetrically:
    x=2kx(y remains the same).x' = 2k - x \quad \text{(y remains the same)}.
  • For each vertex:
    • K(6,4)K(-6, 4)K(1+(1(6))=4,4)K'(-1 + (-1 - (-6)) = 4, 4)K(2,4)K'(-2, 4)
    • L(4,4)L(-4, 4)L(2,4)L'(-2, 4)
    • M(4,2)M(-4, -2)M(2,2)M'(-2, -2)
    • N(6,2)N(-6, -2)N(2,2)N'(-2, -2)

Coordinates of K'L'M'N' after reflection:

  • K(2,4),L(2,4),M(2,2),N(2,2)K'(-2, 4), L'(-2, 4), M'(-2, -2), N'(-2, -2).

Part (c):

Reflection of K'L'M'N' over y=2y = -2:

  • When reflecting a point over y=ky = k, the yy-coordinate changes symmetrically:
    y=2ky(x remains the same).y' = 2k - y \quad \text{(x remains the same)}.
  • For each vertex:
    • K(2,4)K'(-2, 4)K(2,8)K''(-2, -8)
    • L(2,4)L'(-2, 4)L(2,8)L''(-2, -8)
    • M(2,2)M'(-2, -2)M(2,2)M''(-2, -2)
    • N(2,2)N'(-2, -2)N(2,2)N''(-2, -2).

Coordinates of K''L''M''N'' after reflection:

  • K(2,8),L(2,8),M(2,2),N(2,2)K''(-2, -8), L''(-2, -8), M''(-2, -2), N''(-2, -2).

Question 15 (Triangle ABC):

Part (a):

Reflection of ABC\triangle ABC over the yy-axis:

  • When reflecting a point over the yy-axis, the xx-coordinate changes sign:
    (x,y)(x,y).(x, y) \rightarrow (-x, y).
  • For each vertex:
    • A(8,6)A(-8, -6)A(8,6)A'(8, -6)
    • B(3,6)B(-3, -6)B(3,6)B'(3, -6)
    • C(2,3)C(-2, -3)C(2,3)C'(2, -3).

Coordinates of ABCA'B'C':

  • A(8,6),B(3,6),C(2,3)A'(8, -6), B'(3, -6), C'(2, -3).

Part (b):

Rotation of ABC\triangle A'B'C' 180° about the origin:

  • When rotating a point 180° around the origin, both coordinates change sign:
    (x,y)(x,y).(x, y) \rightarrow (-x, -y).
  • For each vertex:
    • A(8,6)A'(8, -6)A(8,6)A''(-8, 6)
    • B(3,6)B'(3, -6)B(3,6)B''(-3, 6)
    • C(2,3)C'(2, -3)C(2,3)C''(-2, 3).

Coordinates of ABCA''B''C'':

  • A(8,6),B(3,6),C(2,3)A''(-8, 6), B''(-3, 6), C''(-2, 3).

Part (c):

Translation of ABC\triangle A''B''C'' under (x,y)(x,y+4)(x, y) \rightarrow (x, y + 4):

  • Add 4 to the yy-coordinates:
    • A(8,6)A''(-8, 6)A(8,10)A'''(-8, 10)
    • B(3,6)B''(-3, 6)B(3,10)B'''(-3, 10)
    • C(2,3)C''(-2, 3)C(2,7)C'''(-2, 7).

Coordinates of ABCA'''B'''C''':

  • A(8,10),B(3,10),C(2,7)A'''(-8, 10), B'''(-3, 10), C'''(-2, 7).

Part (d):

Single transformation mapping ABC\triangle ABC to ABC\triangle A'''B'''C''':

  • A reflection over the yy-axis followed by a rotation of 180° and a translation up by 4 units.
  • Alternatively, the final transformation could be described as a single glide reflection or a composite transformation.

Do you want any detailed explanations for any of these steps?


Related Questions:

  1. How do you determine the line of symmetry for a shape?
  2. How do reflections differ across horizontal and vertical lines?
  3. What is the formula for rotating a point about the origin by 90° or 270°?
  4. What happens to the coordinates of a triangle during a reflection over y=xy = x?
  5. How can translations be represented as vector additions?

Tip:
Reflections preserve the size and shape of a figure, but change its position or orientation depending on the axis of reflection.

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Math Problem Analysis

Mathematical Concepts

Reflections
Rotations
Translations
Coordinate Geometry

Formulas

Reflection over x = k: x' = 2k - x
Reflection over y = k: y' = 2k - y
Reflection over y-axis: (x, y) → (-x, y)
Rotation 180° about origin: (x, y) → (-x, -y)
Translation under (x, y) → (x, y + k)

Theorems

Reflection symmetry
Rotation properties
Translation rules

Suitable Grade Level

Grade 8-10