How to compare side lengths after a dilation on a coordinate plane

Key takeaway: A dilation with scale factor $0.2$ shrinks every length to one-fifth of the original. Once you know that, the equations become much easier to check.

A dilation centered at $(0,0)$ with rule $(0.2x,\ 0.2y)$ has scale factor $0.2$.

That means:

• every image length = $0.2$ × original length
• every original length = $5$ × image length

Check each choice

A. $RV=5R'V'$

• True
• Since $R'V' = 0.2(RV)$, rearranging gives $RV=5R'V'$.

B. $\frac{R'V'}{RV}=\frac{\sqrt{26}}{0.2\sqrt{26}}$

• False
• The left side should be $0.2$, not $\frac{1}{0.2}=5$.

C. $0.2\sqrt{10}\,RT=\sqrt{10}\,R'T'$

• True
• Since $R'T'=0.2RT$, multiplying both sides by $\sqrt{10}$ keeps the equality valid.

D. $RT=0.2R'T'$

• False
• This reverses the dilation relationship.

E. $0.2T'V'=TV$

• False
• The original should be larger: $TV=5T'V'$.

F. $\frac{TV}{T'V'}=\frac{\sqrt{34}}{0.2\sqrt{34}}$

• True
• The ratio simplifies to $\frac{1}{0.2}=5$, which matches the scale factor relationship.

Correct three equations

A, C, F

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Pitfalls the pros know 👇 The most common slip is mixing up image and original lengths. With a dilation factor less than 1, the image gets smaller, so the image-to-original ratio is the scale factor itself. If you flip the ratio, the answer changes immediately.

What if the problem changes? If the dilation were $(2x,2y)$ instead, every image length would be twice the original, and the correct equations would all reverse direction. For a negative scale factor, the figure would also be reflected through the origin, but the length ratios would still use the absolute value of the scale factor.

Tags: dilation, scale factor, coordinate transformation