Question

figure 1 is a rhombus and figure 2 is a rectangle.
neither figure is a square.
which transformation can be used to map figure 1 onto
itself and to map figure 2 onto itself?
a. a rotation of ( 90^{circ} ) clockwise about the center of the
figure
b. a reflection over a line through the center of the
figure that is parallel to one of the sides of the
figure
c. a reflection over one of the diagonals of the figure
d. a rotation of ( 180^{circ} ) about the center of the figure

Explanation:

Step1: Analyze each option for Figure 1 (rhombus)

• Option A: A rotation of \(90^{\circ}\) clockwise about the center of a rhombus (which is not a square) will not map it onto itself. Because for a non - square rhombus, the angles between adjacent sides are not \(90^{\circ}\).
• Option B: A reflection over a line through the center of the figure that is parallel to one of the sides of the rhombus. A rhombus has reflection symmetry over its diagonals, not over a line parallel to its sides.
• Option C: A reflection over one of the diagonals of the rhombus. A rhombus has reflection symmetry over its diagonals. But we need a transformation that can map Figure 1 (rhombus) onto Figure 2 (rectangle). Reflection over a diagonal of the rhombus will not map it to a rectangle.
• Option D: A rotation of \(180^{\circ}\) about the center of the figure.
• For a rhombus, if we rotate it \(180^{\circ}\) about its center, the opposite vertices are swapped.
• For a rectangle, a \(180^{\circ}\) rotation about its center swaps the opposite vertices.

Step2: Analyze for Figure 2 (rectangle)

• A \(180^{\circ}\) rotation about the center of a rectangle maps it onto itself. The mid - point of the line segment joining any two opposite vertices of a rectangle is the center of the rectangle. When we rotate a rectangle \(180^{\circ}\) about its center, each point \((x,y)\) is mapped to \((-x,-y)\) (assuming the center is at the origin of a coordinate system placed at the center of the rectangle), and the rectangle coincides with its original position.

Answer:

D. a rotation of \(180^{\circ}\) about the center of the figure