triangle def was dilated according to the rule ( d_{o, \frac{1}{3}} ) ( (x,y) o left( \frac{1}{3}x, \frac{1}{3}y
ight) ) to create similar triangle def. image of grid with triangles which statements are true? select three options. ( square ) ( angle f ) corresponds to ( angle f ). ( square ) segment ee is parallel to segment ff. ( square ) the distance from point d to the origin is ( \frac{1}{3} ) the distance of point d to the origin. ( square ) the measure of ( angle e ) is ( \frac{1}{3} ) the measure of ( angle e ). ( square ) ( riangle def sim riangle def ).

Question

triangle def was dilated according to the rule ( d_{o, \frac{1}{3}} ) ( (x,y) 	o left( \frac{1}{3}x, \frac{1}{3}y
ight) ) to create similar triangle def. image of grid with triangles which statements are true? select three options. ( square ) ( angle f ) corresponds to ( angle f ). ( square ) segment ee is parallel to segment ff. ( square ) the distance from point d to the origin is ( \frac{1}{3} ) the distance of point d to the origin. ( square ) the measure of ( angle e ) is ( \frac{1}{3} ) the measure of ( angle e ). ( square ) ( 	riangle def sim 	riangle def ).

Accepted Answer

# Brief Explanations:
1. **∠F corresponds to ∠F'**: Dilation preserves angle correspondence, so ∠F (in DEF) corresponds to ∠F' (in D'E'F'). Wait, the option says ∠F corresponds to ∠F (typo? Likely ∠F' but if as is, no. Wait, maybe a typo, but assuming it's ∠F'—no, the option is ∠F corresponds to ∠F. Wait, no, dilation: corresponding angles are equal and correspond. Wait, the original triangle is DEF, dilated to D'E'F', so ∠F (DEF) corresponds to ∠F' (D'E'F'). But the option says ∠F corresponds to ∠F—maybe a typo, but maybe the option is correct as ∠F (original) and ∠F (image)? No, image is F'. Wait, maybe the option has a typo, but let's check others.
2. **Segment EE' parallel to FF'**: Dilation from origin: lines from origin to E and E' are colinear, same with F and F'. So EE' and FF' are both along lines from origin, so they are parallel (or colinear, hence parallel).
3. **Distance from D' to origin is 1/3 of D to origin**: Dilation rule is (x,y)→(1/3 x, 1/3 y), so distance from origin (using distance formula $ \sqrt{(x)^2 + (y)^2} $) for D' is $ \sqrt{(1/3 x)^2 + (1/3 y)^2} = 1/3 \sqrt{x^2 + y^2} $, which is 1/3 of D's distance.
4. **Measure of ∠E' is 1/3 of ∠E**: Dilation preserves angle measures, so ∠E' = ∠E, not 1/3. So this is false.
5. **△DEF ~ △D'E'F'**: Dilation creates similar triangles, so this is true.

Wait, the first option: if it's ∠F (DEF) corresponds to ∠F' (D'E'F'), but the option says ∠F corresponds to ∠F—maybe a typo, but maybe the option is intended to say ∠F corresponds to ∠F' (but written as ∠F). Alternatively, maybe the first option is correct (assuming it's a typo, and ∠F corresponds to ∠F'—but the option says ∠F corresponds to ∠F). Wait, maybe the original triangle's F and the image's F'—but the option says ∠F corresponds to ∠F. Hmm. Let's re-express:

- Dilation preserves angle correspondence: ∠F (DEF) ↔ ∠F' (D'E'F'). If the option has a typo (∠F' instead of ∠F), then it's true. But as written, ∠F corresponds to ∠F—maybe the image's F is F', so maybe the option is miswritten, but let's check the other options.

So the true statements are:
- Segment EE' is parallel to segment FF' (since both are along rays from the origin, so parallel).
- The distance from D' to origin is 1/3 of D to origin (from dilation rule).
- △DEF ~ △D'E'F' (dilation implies similarity).

Wait, and the first option: if ∠F (original) corresponds to ∠F (image)? No, image is F'. But maybe the option is correct (maybe a typo, and it's ∠F'—but as written, ∠F corresponds to ∠F. Maybe the problem has a typo, but assuming the first option is correct (∠F corresponds to ∠F'—but written as ∠F), or maybe the first option is incorrect. Wait, let's re-evaluate:

1. ∠F corresponds to ∠F: In dilation, corresponding angles are equal and correspond to the image angle. So ∠F (DEF) corresponds to ∠F' (D'E'F'), not ∠F. So this is false? But maybe the option is written as ∠F (image) corresponds to ∠F (original)? No, the notation: D'E'F' is the image, so F' is the image of F. So ∠F (original) corresponds to ∠F' (image). So the first option is false (as written). Then the three true are:

- Segment EE' || FF' (true, colinear with origin, so parallel).
- Distance D' to origin is 1/3 of D to origin (true, from dilation rule).
- △DEF ~ △D'E'F' (true, dilation gives similarity).

Wait, but the first option: maybe the problem has a typo, and it's ∠F corresponds to ∠F'—then it's true. But as written, ∠F corresponds to ∠F—so maybe the first option is incorrect. Let's check each:

- ∠F corresponds to ∠F: False (should be ∠F').
- EE' || FF': True (both on lines through origin, so parallel).
- Distance D' to origin: 1/3 of D to origin: True (dilation scale factor 1/3, distance scales by 1/3).
- ∠E' is 1/3 of ∠E: False (angles preserved).
- △DEF ~ △D'E'F': True (dilation implies similarity).

So the three true are: Segment EE' is parallel to segment FF', The distance from D' to origin is 1/3 of D to origin, △DEF ~ △D'E'F'. Wait, but the first option: maybe the problem meant ∠F corresponds to ∠F'—if so, then that's also true. But as written, ∠F corresponds to ∠F—so maybe the first option is a typo. Assuming the first option is correct (∠F corresponds to ∠F'—but written as ∠F), then the three are: ∠F corresponds to ∠F, Segment EE' || FF', Distance D' to origin, and △DEF ~ △D'E'F'—but we need three. Wait, maybe the first option is correct (∠F in original and ∠F in image? No, image is F'). This is confusing. Let's go with the dilation properties:

- Dilation preserves angle measures and correspondence (so ∠F ↔ ∠F', angles equal).
- Dilation scales distances from the center (origin) by the scale factor (1/3), so distance from D' to origin is 1/3 of D to origin.
- Dilation creates similar triangles, so △DEF ~ △D'E'F'.
- Lines from the center to corresponding points are colinear, so EE' and FF' are both along lines from origin, hence parallel.

So the true statements are:

1. ∠F corresponds to ∠F' (assuming the option has a typo, and it's ∠F' instead of ∠F—otherwise, false). But if the option is written as ∠F corresponds to ∠F, maybe it's a mistake, but let's check the options again. The first option: "∠F corresponds to ∠F"—maybe it's a typo, and it's ∠F'—so we'll take it as true. Then the three are:

- ∠F corresponds to ∠F (assuming typo, ∠F'—true).
- Segment EE' is parallel to segment FF' (true).
- The distance from D' to origin is 1/3 of D to origin (true).
- △DEF ~ △D'E'F' (true).

But we need three. Wait, the problem says "select three options". Let's confirm:

- ∠F corresponds to ∠F: If it's ∠F (original) and ∠F (image), no—image is F'. So this is false. Then the three true are:

- Segment EE' || FF' (true).
- Distance D' to origin (true).
- △DEF ~ △D'E'F' (true).

Yes, these three. The first option is false (typo), the fourth is false (angles preserved), so the three are:

- Segment EE' is parallel to segment FF'.
- The distance from point D' to the origin is $ \frac{1}{3} $ the distance of point D to the origin.
- $ \triangle DEF \sim \triangle D'E'F' $.

# Answer:
- ∠F corresponds to ∠F: (Assuming typo, but as per options, if it's ∠F'—but written as ∠F, maybe incorrect. However, the correct three are:
  - Segment EE' is parallel to segment FF'.
  - The distance from point D' to the origin is $ \frac{1}{3} $ the distance of point D to the origin.
  - $ \triangle DEF \sim \triangle D'E'F' $.

(Note: If the first option is intended to be ∠F corresponds to ∠F' (image), then it's also true. But based on the written option, it's ∠F corresponds to ∠F, which is likely a typo. The three definite true are the last three except the fourth.)

So the three correct options are:
- Segment EE' is parallel to segment FF'.
- The distance from point D' to the origin is $ \frac{1}{3} $ the distance of point D to the origin.
- $ \triangle DEF \sim \triangle D'E'F' $.

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