Math Problem Statement
Solution
Here are the solutions to the questions provided:
Question 14 (Parallelogram KLMN):
Part (a):
- Draw the lines (a vertical line) and (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 :
- When reflecting a point over a vertical line , the -coordinate changes symmetrically:
- For each vertex:
- → →
- →
- →
- →
Coordinates of K'L'M'N' after reflection:
- .
Part (c):
Reflection of K'L'M'N' over :
- When reflecting a point over , the -coordinate changes symmetrically:
- For each vertex:
- →
- →
- →
- → .
Coordinates of K''L''M''N'' after reflection:
- .
Question 15 (Triangle ABC):
Part (a):
Reflection of over the -axis:
- When reflecting a point over the -axis, the -coordinate changes sign:
- For each vertex:
- →
- →
- → .
Coordinates of :
- .
Part (b):
Rotation of 180° about the origin:
- When rotating a point 180° around the origin, both coordinates change sign:
- For each vertex:
- →
- →
- → .
Coordinates of :
- .
Part (c):
Translation of under :
- Add 4 to the -coordinates:
- →
- →
- → .
Coordinates of :
- .
Part (d):
Single transformation mapping to :
- A reflection over the -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:
- How do you determine the line of symmetry for a shape?
- How do reflections differ across horizontal and vertical lines?
- What is the formula for rotating a point about the origin by 90° or 270°?
- What happens to the coordinates of a triangle during a reflection over ?
- 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
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