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Rectangle [tex]$ABCD$[/tex] with coordinates [tex]$A(1,1), B(4,1), C(4,2)$[/tex], and [tex]$D(1,2)$[/tex] dilates with respect to the origin to give rectangle [tex]$A'B'C'D'$[/tex] such that [tex]$\overline{A'B'}=6$[/tex]. What is the scale factor of the dilation?

A. 6
B. 3
C. 2
D. [tex]$\frac{\pi}{2}$[/tex]
E. 4

Sagot :

GinnyAnsweridn
To determine the scale factor of the dilation, let's follow these steps methodically.

1. Determine the initial length of side [tex]\( \overline{AB} \)[/tex] of rectangle [tex]\( ABCD \)[/tex]:

The coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are [tex]\( A(1, 1) \)[/tex] and [tex]\( B(4, 1) \)[/tex], respectively. The length of [tex]\( \overline{AB} \)[/tex] is the horizontal distance between these two points.

[tex]\[ \text{Length of } \overline{AB} = B_x - A_x = 4 - 1 = 3 \][/tex]

2. Find the length of the transformed side [tex]\( \overline{A'B'} \)[/tex] of rectangle [tex]\( A'B'C'D' \)[/tex]:

The problem states that the length of [tex]\( \overline{A'B'} = 6 \)[/tex].

3. Calculate the scale factor of the dilation:

The scale factor of a dilation is the ratio of the length of the transformed side to the length of the original side. Therefore,

[tex]\[ \text{Scale factor} = \frac{\text{Length of } \overline{A'B'}}{\text{Length of } \overline{AB}} = \frac{6}{3} = 2 \][/tex]

So, the scale factor of the dilation is [tex]\( 2 \)[/tex].

Thus, the correct answer is:

C. 2
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