therefore, the quotients of the corresponding side lengths of the figures show that figure jklm is not a dilation of figure pqrs because all of the quotients are equivalent.

Question

Accepted Answer

To determine the correct terms, we recall the definition of dilation: a dilation is a transformation that produces an image that is the same shape as the original, but a different size. For a dilation, the ratios (quotients) of corresponding side lengths must all be equal (the scale factor).

### Analyzing the first blank:
If the quotients of corresponding side lengths are equivalent, then the figure **is** a dilation. Wait, but the original statement says "the quotients... show that figure JKLM [blank] a dilation of figure PQRS because [blank] of the quotients are equivalent." Wait, no—wait, maybe a typo? Wait, no: if *all* quotients are equivalent, then it **is** a dilation. Wait, but the first blank is "is not" or the opposite? Wait, no—let's re-express.

Wait, the key: Dilation requires that **all** corresponding side length ratios (quotients) are equal (the scale factor). So if "all of the quotients are equivalent," then the figure **is** a dilation. But the original first blank is "is not"—that would be incorrect. Wait, maybe the original problem had a different context, but based on dilation definition:

- For a dilation, the ratios of corresponding sides must be equal (i.e., *all* quotients are equivalent).
- So if "all of the quotients are equivalent," then figure JKLM **is** a dilation of PQRS. But the first blank is "is not"—that would be wrong. Wait, maybe the original problem had a mistake, but let's check the logic.

Wait, the sentence structure: "the quotients... show that figure JKLM [blank] a dilation of figure PQRS because [blank] of the quotients are equivalent."

If "all" quotients are equivalent, then it **is** a dilation. So the first blank should be "is", and the second blank "all". But the original first blank is "is not"—maybe a misstatement. Wait, no—maybe the problem is that if *not all* quotients are equivalent, then it is not a dilation. But the sentence says "because [blank] of the quotients are equivalent". So if "all" are equivalent, then it **is** a dilation.

But let's correct based on dilation definition:

1. First blank: If all quotients are equivalent, then JKLM **is** a dilation. But the original has "is not"—maybe that's an error. Wait, no—maybe the problem is phrased as "is not" because maybe the quotients *are not all* equivalent? Wait, the sentence says "because [blank] of the quotients are equivalent". So if "all" are equivalent, then it is a dilation. So the correct first blank is "is", and second blank "all". But the original first blank is "is not"—maybe a typo. Alternatively, maybe the intended answer is:

Wait, the user's problem shows "is not" and "all" as options. Wait, no—let's re-express.

Wait, the key: Dilation requires that the ratio of corresponding sides is constant (all quotients equal). So:

- If "all" quotients are equivalent, then JKLM **is** a dilation.
- If "not all" quotients are equivalent, then JKLM **is not** a dilation.

But the sentence says "because [blank] of the quotients are equivalent". So if the reason is that "all" are equivalent, then the first blank should be "is". But the original first blank is "is not"—this is confusing. Wait, maybe the problem has a mistake, but based on the options (the dropdowns: first is "is not" or maybe "is", second is "all" or "not all").

Wait, let's assume the intended logic:

To be a dilation, *all* corresponding side length quotients must be equal. So:

- If "all" quotients are equivalent, then JKLM **is** a dilation.
- If "not all" quotients are equivalent, then JKLM **is not** a dilation.

But the sentence is: "the quotients... show that figure JKLM [blank] a dilation of figure PQRS because [blank] of the quotients are equivalent."

So if the "because" clause is "all of the quotients are equivalent", then the first blank is "is". If the "because" clause were "not all", then first blank is "is not".

But the original problem's second blank is "all" (as per the dropdown). So let's check:

If we use "is" and "all":

"Therefore, the quotients of the corresponding side lengths of the figures show that figure JKLM **is** a dilation of figure PQRS because **all** of the quotients are equivalent."

This matches the dilation definition.

But the original first blank is "is not"—maybe that's an error. Alternatively, maybe the problem intended to say "is not" because the quotients are *not* all equivalent, but the "because" clause is "all"—that would be contradictory.

Wait, perhaps the user made a typo, but based on dilation definition, the correct answers are:

First blank: "is" (because all quotients equivalent implies dilation)
Second blank: "all"

But if we follow the original dropdown (first is "is not", second is "all"), that would be incorrect. Wait, no—maybe the problem is reversed. Let's think again:

Wait, maybe the figures are not a dilation, so the first blank is "is not", and the second blank is "not all"—but the original second blank is "all". This is confusing.

Wait, perhaps the intended answer is:

First blank: "is"
Second blank: "all"

Because dilation requires all corresponding side ratios to be equal.

So correcting the sentence:

"Therefore, the quotients of the corresponding side lengths of the figures show that figure JKLM **is** a dilation of figure PQRS because **all** of the quotients are equivalent."

# Brief Explanations:
Dilation requires the ratio (quotient) of corresponding side lengths to be constant (all quotients equal). If all quotients are equivalent, the figure is a dilation.

# Answer:
First blank: is
Second blank: all

(Note: If the original problem’s first blank is forced to "is not", it would be incorrect, as "all quotients equivalent" implies dilation. The correct logical pairing is "is" (first blank) and "all" (second blank) to satisfy the dilation definition.)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Analyzing the first blank:

Answer: