Question

what is true about the image △qrs? select three options. which statements are true? □d_{o,2}(x,y)=(2x,2y) □side qs lies on a line with a slope of -1. □overline{qr} is longer than overline{qr}. □the vertices of the image are closer to the origin than those of the pre - image. □the distance from q to the origin is twice the distance from q to the origin.

Explanation:

Step1: Recall dilation properties

A dilation with a scale - factor of \(k = 2\) (from \(D_{O,2}\)) multiplies the coordinates of each point by \(2\). If a point \((x,y)\) is dilated by a scale - factor \(k\) centered at the origin \((0,0)\), the new point is \((kx,ky)\).

Step2: Analyze each statement

Statement 1: \(D_{O,2}(x,y)=(2x,2y)\)

This is the correct rule for a dilation of scale - factor \(2\) centered at the origin. When we dilate a point \((x,y)\) by a scale - factor of \(2\) centered at the origin, we multiply the \(x\) and \(y\) coordinates by \(2\).

Statement 2: Side \(Q'S'\) lies on a line with a slope of \(- 1\)

First, find the coordinates of the pre - image points. Let \(Q(-3, - 1)\), \(R(2,4)\), \(S(1, - 1)\). After dilation \(Q'(-6,-2)\), \(R'(4,8)\), \(S'(2,-2)\). The slope of the line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \(Q'(-6,-2)\) and \(S'(2,-2)\), \(m=\frac{-2-(-2)}{2 - (-6)}=0
eq - 1\).

Statement 3: \(\overline{Q'R'}\) is longer than \(\overline{QR}\)

The length of a line segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). If we dilate a line segment by a scale - factor \(k\), the new length is \(k\) times the original length. Since \(k = 2\), the length of \(\overline{Q'R'}=2\times\overline{QR}\), so \(\overline{Q'R'}\) is longer than \(\overline{QR}\).

Statement 4: The vertices of the image are closer to the origin than those of the pre - image

Since the scale - factor \(k = 2\), the distance of each vertex from the origin in the image is \(2\) times the distance of the corresponding vertex from the origin in the pre - image. So the vertices of the image are farther from the origin.

Statement 5: The distance from \(Q'\) to the origin is twice the distance from \(Q\) to the origin

The distance of a point \((x,y)\) from the origin is \(d=\sqrt{x^{2}+y^{2}}\). If \(Q(x_1,y_1)\) and \(Q'(2x_1,2y_1)\), the distance of \(Q\) from the origin is \(d_1=\sqrt{x_1^{2}+y_1^{2}}\) and the distance of \(Q'\) from the origin is \(d_2=\sqrt{(2x_1)^{2}+(2y_1)^{2}}=\sqrt{4x_1^{2}+4y_1^{2}} = 2\sqrt{x_1^{2}+y_1^{2}}\). So the distance from \(Q'\) to the origin is twice the distance from \(Q\) to the origin.

Answer:

\(D_{O,2}(x,y)=(2x,2y)\) is true.
\(\overline{Q'R'}\) is longer than \(\overline{QR}\) is true.
The distance from \(Q'\) to the origin is twice the distance from \(Q\) to the origin is true.