Question

7 parallelograms fghj and wxyz are shown on the coordinate plane. which sequences of transformations can be used to show similarity between the parallelograms? select the two correct sequences of transformations. a a rotation 270° clockwise about the origin followed by a dilation centered at the origin with scale factor 1/2 b a dilation centered at the origin with scale factor 1/2 followed by a rotation 180° clockwise about the origin c a dilation centered at the origin with scale factor 1/2 followed by a rotation 90° counterclockwise about the origin d a rotation 90° counterclockwise about the origin followed by a translation down 2 units e a reflection over the y - axis followed by a dilation centered at the origin with scale factor 1/2 f a translation down 2 units followed by a reflection over the y axis

Explanation:

Step1: Recall the properties of similarity transformations

Similarity transformations include rotations, dilations, reflections. A dilation changes the size by a scale factor, and rotations/reflections change the orientation.

Step2: Analyze each option

• Option A:

A \(270^{\circ}\) clock - wise rotation about the origin: The rule for a \(270^{\circ}\) clock - wise rotation \((x,y)\to(y, - x)\). Then a dilation with scale factor \(\frac{1}{2}\): \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\).

• Option B:

A dilation with scale factor \(\frac{1}{2}\): \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\). Then a \(180^{\circ}\) clock - wise rotation: \((x,y)\to(-x, - y)\).

• Option C:

A dilation with scale factor \(\frac{1}{2}\): \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\). Then a \(90^{\circ}\) counter - clockwise rotation: \((x,y)\to(-y,x)\).

• Option D:

A \(90^{\circ}\) counter - clockwise rotation: \((x,y)\to(-y,x)\). Then a translation down \(2\) units: \((x,y)\to(x,y - 2)\). Translation and rotation (without dilation) do not change the size proportionally (since parallelograms are of different sizes, we need a dilation for similarity).

• Option E:

A reflection over the \(y\) - axis: \((x,y)\to(-x,y)\). Then a dilation with scale factor \(\frac{1}{2}\): \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\). But the orientation is not correct.

• Option F:

A translation down \(2\) units: \((x,y)\to(x,y - 2)\). Then a reflection over the \(y\) - axis: \((x,y)\to(-x,y)\). Translation and reflection (without dilation) do not change the size proportionally.

Let's assume a point. Take a vertex of parallelogram \(FGHJ\) say \(F(0,2)\), \(G(2,6)\), \(J(6,2)\), \(H(8,6)\). For parallelogram \(WXYZ\), \(W(0,0)\), \(X(- 4,1)\), \(Y(-4,4)\), \(Z(0,3)\) (approximate coordinates from the graph).
If we consider the scale factor. The length of \(FJ\) (in \(FGHJ\)): Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(F(0,2)\) and \(J(6,2)\), \(d=\sqrt{(6 - 0)^2+(2 - 2)^2}=6\). For \(WX\) (in \(WXYZ\)), using \(W(0,0)\) and \(X(-4,1)\), \(d=\sqrt{(-4 - 0)^2+(1 - 0)^2}=\sqrt{17}\approx4.12\) (not relevant for similarity in terms of just translation/rotation). But if we use dilation.
If we consider the ratio of side lengths. The side lengths of \(FGHJ\) are larger. A dilation with scale factor \(\frac{1}{2}\) is needed.
If we use rotation:

• For a \(270^{\circ}\) clock - wise rotation: \((x,y)\to(y, - x)\). Then dilation \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)
• For a dilation \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\) first and then \(180^{\circ}\) rotation \((x,y)\to(-x, - y)\)

Answer:

A. a rotation \(270^{\circ}\) clockwise about the origin followed by a dilation centered at the origin with scale factor \(\frac{1}{2}\)
B. a dilation centered at the origin with scale factor \(\frac{1}{2}\) followed by a rotation \(180^{\circ}\) clockwise about the origin