12 multiple-choice questions, progressively harder.
A dilation has center O(1,−2)O(1,-2)O(1,−2) and scale factor 12\frac 1221. Find the image of P(5,4)P(5, 4)P(5,4).
Solution
Correct answer: C
From the center to P, the displacement is four right and six up. Halve it and add the center back.
A dilation has scale factor 1.81.81.8. The image of a segment has length 12.612.612.6 cm. What is the original segment length?
Correct answer: A
The image length equals the factor times the original length. Undo the multiplication.
L=12.61.8,=7 cm.\begin{aligned}L&=\frac{12.6}{1.8},\\&=7\text{ cm}.\end{aligned}L=1.812.6,=7 cm.
A dilation centered at the origin has scale factor 34\frac 3443. What is the image of the horizontal line y=4y=4y=4?
Correct answer: D
Every point on the original line has vertical coordinate four. Its image vertical coordinate is therefore three.
y′=34⋅4,=3.\begin{aligned}y'&=\frac 34\cdot 4,\\&=3.\end{aligned}y′=43⋅4,=3.
The image horizontal line contains all real horizontal coordinates, so it is the entire line y = 3.
A dilation has center (−2,3)(-2, 3)(−2,3) and scale factor 1.51.51.5. Where does (2,−1)(2,-1)(2,−1) map?
Correct answer: B
Scale the displacements from the center, not the raw point coordinates.
Line mmm passes through a dilation's center OOO. The factor is 222. Which statement is correct?
The two rays of the line from O map onto themselves. Individual positive distances double, so only O stays in its original location.
An angle has measure 75∘75^\circ75∘. Its image under a dilation is labeled (2x+17)∘(2x+17)^\circ(2x+17)∘. Find xxx.
Angle measures stay equal under a dilation.
Points O,P,P′O,P,P'O,P,P′ occur in that order on one ray. A dilation centered at OOO sends PPP to P′P'P′. If
, what is its scale factor?
First add the two adjacent distances to get the distance from O to the image.
Points O,P′,PO,P',PO,P′,P occur in that order on one ray. A dilation centered at OOO sends PPP to P′P'P′. If
The whole original distance includes both adjacent pieces. Divide the image distance by that whole distance.
A segment of length 2t+22t+22t+2 has an image of length 4t−24t-24t−2 under a dilation with scale factor 1.51.51.5. Find ttt.
The entire original expression is multiplied by the factor.
This gives positive lengths 12 and 18, whose image-to-original ratio is 1.5.
A dilation has center (2,−1)(2,-1)(2,−1) and scale factor 333. What is the image of the line y=−1y=-1y=−1?
The center lies on the line y = -1. A line through the center maps onto the same line as a set, even though most of its points move.
Which coordinate rule gives a dilation centered at the origin with scale factor 14\frac 1441?
Multiply both coordinates by the same factor. Dividing by four is multiplying by one quarter.
A dilation with scale factor 333 sends (2,1)(2, 1)(2,1) to (8,7)(8, 7)(8,7). What is its center?
Let the center be (a,b) and use the two displacement rules.
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