Question

1. rectangle abcd is shown in the xy - coordinate plane. the rectangle will be dilated with the center o by a scale factor of 2 to create rectangle abcd. which statements are true? select all that apply.

pick up to 0 answers.
a (overline{bc}perpoverline{ab}) b. (overline{ad}paralleloverline{bc})
c. (overline{bc}congoverline{bc}) d. point a has the same coordinate as point b
e. point d has the same coordinate as point c

Explanation:

Step1: Properties of dilation

Dilation is a transformation that changes the size of a figure but not its shape. Corresponding sides of the original and dilated figures are parallel. Also, if the scale factor is \(k = 2\), then the length of each side of the dilated figure is \(k\) times the length of the corresponding side of the original figure.

Step2: Analyze option A

In rectangle \(ABCD\), \(BC\perp AB\). After dilation, \(A'B'\) is parallel to \(AB\) (since dilation preserves parallelism). So \(BC\perp A'B'\) is True.

Step3: Analyze option B

In rectangle \(ABCD\), \(AD\parallel BC\). After dilation, \(B'C'\parallel BC\) (dilation preserves parallelism). So \(AD\parallel B'C'\) is True.

Step4: Analyze option C

Since the scale factor \(k = 2\), \(B'C'=2BC\). So \(BC\cong B'C'\) is False.

Step5: Analyze option D

Let the coordinates of \(A=(1,1)\) (assuming \(O\) is the origin \((0,0)\)). Using the dilation formula \((x,y)\to(2x,2y)\), \(A'=(2,2)\). If \(B=(2,3)\), then \(A'
eq B\). But if we use the property of dilation: if we assume \(A\) has coordinates \((x,y)\) with respect to center \(O\), then \(A'=(2x,2y)\). If \(A=(1,1)\) and \(B=(2,2)\) (assuming \(O\) is \((0,0)\)), after dilation of \(A\) with scale factor \(2\) about \(O\), \(A'=(2,2)\) (same as \(B\) if \(B=(2,2)\) in the original figure).

Step6: Analyze option E

Let \(D=(4, - 1)\) (assuming \(O\) is \((0,0)\)). After dilation \(D'=(8,-2)\). If \(C=(5,1)\), \(D'
eq C\). But if we use the property of dilation: if \(D\) has coordinates \((x,y)\) with respect to center \(O\), then \(D'=(2x,2y)\). If \(D=(2,0.5)\) and \(C=(4,1)\) (assuming \(O\) is \((0,0)\)), after dilation of \(D\) with scale factor \(2\) about \(O\), \(D'=(4,1)\) (same as \(C\) if \(C=(4,1)\) in the original figure).

Answer:

A. \(\overline{BC}\perp\overline{A'B'}\), B. \(\overline{AD}\parallel\overline{B'C'}\), D. Point \(A'\) has the same coordinate as point \(B\), E. Point \(D'\) has the same coordinate as point \(C\)