Question

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unit 2, lesson 5 cool down: criss cross
what happens to the diagonal of a rectangle when the rectangle is reflected across a line of symmetry?
what does this suggest about the diagonals of rectangles?

unit 2, lesson 6 cool down: mystery quad
quadrilateral abcd has both reflection and rotation symmetry. what type of quadrilateral could abcd be? show or explain your reasoning.

Explanation:

Step1: Analyze rectangle diagonal reflection

When a rectangle is reflected across a line of symmetry, the diagonal is mapped onto the other diagonal. This is because the line of symmetry of a rectangle bisects the rectangle into two congruent halves, and the diagonals are symmetrically placed with respect to this line.

Step2: Infer diagonal property of rectangles

This suggests that the diagonals of a rectangle are congruent. Since reflection is a rigid - motion that preserves length, and one diagonal can be mapped onto the other by reflection across a line of symmetry, their lengths must be equal.

Step3: Analyze quadrilateral with both symmetries

For a quadrilateral \(ABCD\) with both reflection and rotation symmetry: A square has both reflection symmetry (it has 4 lines of symmetry) and rotation symmetry (it can be rotated by 90°, 180°, 270° about its center and still look the same). A rectangle also has 2 lines of symmetry and 180° rotation symmetry about its center. A rhombus has 2 lines of symmetry and 180° rotation symmetry about its center. So, \(ABCD\) could be a square, a rectangle, or a rhombus.

Answer:

• The diagonal of a rectangle is mapped onto the other diagonal when reflected across a line of symmetry.
• The diagonals of rectangles are congruent.
• The quadrilateral \(ABCD\) could be a square, a rectangle, or a rhombus.