Math  /  Geometry

Question1. A reflection over the xx-axis maps ABC\triangle A B C to ABC\triangle A^{\prime} B^{\prime} C^{\prime}. Do the preimage and image have the same size and shape? Explain. Find a congruence transformation that maps RST\triangle R S T to UVW\triangle U V W. 2. 3.

Studdy Solution

STEP 1

1. A reflection over the xx-axis is an isometry, meaning it preserves distances and angles.
2. Congruence transformations include reflections, rotations, and translations.

STEP 2

1. Determine if the preimage and image of a reflection over the xx-axis have the same size and shape.
2. Identify a congruence transformation for RST\triangle RST to UVW\triangle UVW in the first grid.
3. Identify a congruence transformation for RST\triangle RST to UVW\triangle UVW in the second grid.

STEP 3

A reflection over the xx-axis maps each point (x,y)(x, y) to (x,y)(x, -y). This transformation preserves the size and shape of the figure, as it is an isometry.
Conclusion: Yes, the preimage ABC\triangle ABC and the image ABC\triangle A'B'C' have the same size and shape because a reflection is a congruence transformation.

STEP 4

To find a congruence transformation that maps RST\triangle RST to UVW\triangle UVW in the first grid, observe the coordinates:
- Reflect RST\triangle RST over the xx-axis: - R(5,1)R(5,1)R(-5, 1) \rightarrow R'(-5, -1) - S(2,4)S(2,4)S(-2, 4) \rightarrow S'(-2, -4) - T(1,2)T(1,2)T(1, 2) \rightarrow T'(1, -2)
- Translate RST\triangle R'S'T' down by 1 unit: - R(5,1)U(5,2)R'(-5, -1) \rightarrow U(-5, -2) - S(2,4)V(2,5)S'(-2, -4) \rightarrow V(-2, -5) - T(1,2)W(1,3)T'(1, -2) \rightarrow W(1, -3)
Conclusion: The congruence transformation is a reflection over the xx-axis followed by a translation down by 1 unit.

STEP 5

To find a congruence transformation that maps RST\triangle RST to UVW\triangle UVW in the second grid, observe the coordinates:
- Rotate RST\triangle RST 9090^\circ counterclockwise around the origin: - R(1,5)R(5,1)R(1, -5) \rightarrow R'(5, 1) - S(4,2)S(2,4)S(4, -2) \rightarrow S'(2, 4) - T(2,1)T(1,2)T(2, 1) \rightarrow T'(-1, 2)
- Translate RST\triangle R'S'T' by vector (0,1)(0, 1): - R(5,1)U(5,2)R'(5, 1) \rightarrow U(5, 2) - S(2,4)V(2,5)S'(2, 4) \rightarrow V(2, 5) - T(1,2)W(1,3)T'(-1, 2) \rightarrow W(-1, 3)
Conclusion: The congruence transformation is a 9090^\circ counterclockwise rotation around the origin followed by a translation up by 1 unit.

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