Question

quadrilaterals 1, 2, and 3 are shown on the coordinate plane.
which statements can be used to show similarity between the quadrilaterals?
select the two correct statements.
a a dilation with scale factor \\( \frac { 1 } { 3 } \\) centered at the origin maps quadrilateral 1 onto quadrilateral 2.
b a translation down 4 units followed by a translation right 2 units maps quadrilateral 1 onto quadrilateral 2.
c a rotation \\( 180 ^ { \circ } \\) about the origin maps quadrilateral 2 onto quadrilateral 3.
d a rotation \\( 180 ^ { \circ } \\) about the origin maps quadrilateral 1 onto quadrilateral 3.
e a translation up 2 units followed by a translation right 5 units maps quadrilateral 2 onto quadrilateral 3.
f a dilation with scale factor 3 centered at the origin maps quadrilateral 1 onto quadrilateral 2.

Explanation:

Step1: Analyze dilation

Dilation changes the size of a figure. If we consider the side - length ratios. For a dilation centered at the origin, if the scale factor is \(k=\frac{1}{3}\), the coordinates of the vertices of the dilated figure \((x,y)\) from the original figure \((x_0,y_0)\) are given by \((x = kx_0,y=ky_0)\).

Step2: Analyze rotation

A rotation of \(180^{\circ}\) about the origin transforms a point \((x,y)\) to \((-x, - y)\).

Step3: Analyze translation

Translation is a slide. A translation \((x,y)\to(x + a,y + b)\) moves the figure \(a\) units horizontally and \(b\) units vertically.

Let's check each option:

• Option A:

If we assume a dilation with scale factor \(\frac{1}{3}\) centered at the origin. Let's take a vertex of quadrilateral 1, say \((- 6,6)\). After dilation with \(k = \frac{1}{3}\), the new vertex is \((-2,2)\). If we check the vertices of quadrilateral 2, we can see that a dilation with scale factor \(\frac{1}{3}\) centered at the origin maps quadrilateral 1 onto quadrilateral 2.

• Option B:

A translation is a rigid - motion (preserves size). Quadrilateral 1 and 2 are not congruent (because of the size difference), so a translation cannot map 1 onto 2.

• Option C:

A rotation of \(180^{\circ}\) about the origin. Let's take a vertex of quadrilateral 2, say \((-1,-1)\). After rotation \(180^{\circ}\) about the origin \((x,y)\to(-x,-y)\), we get \((1,1)\). But the vertices of quadrilateral 3 do not match the result of rotating quadrilateral 2 by \(180^{\circ}\) about the origin.

• Option D:

Let's take a vertex of quadrilateral 1, say \((-6,6)\). After rotation \(180^{\circ}\) about the origin \((x,y)\to(-x,-y)\), we get \((6,-6)\). But the vertices of quadrilateral 3 do not match the result of rotating quadrilateral 1 by \(180^{\circ}\) about the origin.

• Option E:

A translation is a rigid - motion (preserves size). Quadrilateral 2 and 3 are not congruent (because of the size difference), so a translation cannot map 2 onto 3.

• Option F:

If we assume a dilation with scale factor 3 centered at the origin. Let's take a vertex of quadrilateral 2, say \((-2,2)\). After dilation with \(k = 3\), the new vertex is \((-6,6)\) (a vertex of quadrilateral 1), but we want to map 1 onto 2. The scale factor should be \(\frac{1}{3}\) (not 3) for mapping 1 onto 2.

Answer:

A. A dilation with scale factor \(\frac{1}{3}\) centered at the origin maps quadrilateral 1 onto quadrilateral 2.