Question

which statements justify that the dilation of triangle stu is an enlargement increasing its size by the magnitude of the scale factor? choose two correct answers. the triangle pre - image is closer to the point of dilation than the image. the value of the scale factor, \\(\frac{1}{6}\\), is between 0 and 1. the vertices of the pre - image are collinear with the vertices of the image. the value of the scale factor, 6, is greater than 1. the triangle image is closer to the point of dilation than the pre - image.

Explanation:

Step1: Understand Dilation Properties

Dilation is a transformation that changes the size of a figure. The scale factor \( k \) determines if it's an enlargement (\( k>1 \)) or reduction (\( 0<k<1 \)). Collinearity: Pre - image and image vertices lie on lines through the center of dilation (point \( P \)). Distance from dilation center: For enlargement (\( k > 1 \)), image is farther from \( P \) than pre - image; for reduction (\( 0<k < 1 \)), pre - image is farther.

Step2: Analyze Each Statement

1. **"The triangle pre - image is closer to the point of dilation than the image."**
For an enlargement (\( k>1 \)), the image is farther from the center (\( P \)) than the pre - image. So the pre - image is closer? No, wait: If \( k>1 \), image is larger and farther. Wait, no: Let's calculate scale factor. From the diagram, \( PT = 2 \), \( PT'=10 \). Scale factor \( k=\frac{PT'}{PT}=\frac{10}{2} = 5\)? Wait, maybe I misread. Wait, the pre - image is the smaller triangle (STU), image is \( S'T'U' \). The distance from \( P \) to pre - image vertices is less than to image vertices? No, wait: If it's an enlargement, the image is farther. Wait, maybe the first statement is wrong. Wait, no, let's re - check. Wait, the statement says "pre - image is closer to dilation point than image". For enlargement (\( k > 1 \)), image is farther, so pre - image is closer. Wait, maybe I made a mistake. Wait, let's take the scale factor. Let's say the length from \( P \) to \( T \) is \( 2 \), to \( T' \) is \( 10 \). So scale factor \( k=\frac{10}{2}=5 \), which is \( k > 1 \) (enlargement). So the image (\( S'T'U' \)) is farther from \( P \) than the pre - image (\( STU \)). So the pre - image is closer to \( P \) than the image? Wait, no: If \( PT = 2 \), \( PT'=10 \), then the pre - image vertex \( T \) is at distance \( 2 \) from \( P \), image vertex \( T' \) is at \( 10 \) from \( P \). So pre - image is closer. So this statement is correct? Wait, no, the first statement says "The triangle pre - image is closer to the point of dilation than the image." So if \( k>1 \), image is farther, so pre - image is closer. So this statement is correct? Wait, maybe I messed up. Wait, no, let's check the second statement: "The value of the scale factor, \( \frac{1}{6} \), is between 0 and 1." But \( \frac{1}{6}<1 \), which is a reduction, not enlargement. So this is wrong.

Third statement: "The vertices of the pre - image are collinear with the vertices of the image."
By the definition of dilation, all corresponding vertices (and the center \( P \)) are collinear. So this is correct.

Fourth statement: "The value of the scale factor, 6, is greater than 1."
If we calculate scale factor as \( \frac{\text{image length}}{\text{pre - image length}} \), from \( PT = 2 \), \( PT'=10 \), \( k=\frac{10}{2}=5 \), but 6 is greater than 1, and since enlargement has \( k > 1 \), the fact that scale factor is greater than 1 is true (even if the actual scale factor is 5, the statement says "the value of the scale factor, 6, is greater than 1" – the key is that for enlargement, scale factor \( k>1 \), so this statement's logic is correct (since 6 > 1, and enlargement requires \( k>1 \)).

Fifth statement: "The triangle image is closer to the point of dilation than the pre - image."
No, as we saw, image is farther, so this is wrong.

Wait, maybe I made a mistake in the first statement. Let's re - evaluate. The problem says "the dilation of triangle STU is an enlargement (increasing its size)". So \( k>1 \). So the image is larger and farther from \( P \) than the pre - image. So the pre - image is closer to \( P \) than the image. So the first statement: "The triangle pre - image is closer to the point of dilation than the image." is correct.

The third statement: "The vertices of the pre - image are collinear with the vertices of the image." is correct (by dilation property: center, pre - image vertex, image vertex are collinear).

The fourth statement: "The value of the scale factor, 6, is greater than 1." Since enlargement requires \( k > 1 \), and 6>1, this is correct (even if the actual scale factor is, say, 5, the statement is about the nature of the scale factor for enlargement, i.e., \( k>1 \)).

Wait, but the problem says "choose two correct answers". Let's re - check:

- **"The triangle pre - image is closer to the point of dilation than the image."** Correct (enlargement: image is farther, pre - image closer).
- **"The value of the scale factor, \( \frac{1}{6} \), is between 0 and 1."** Wrong (reduction, not enlargement).
- **"The vertices of the pre - image are collinear with the vertices of the image."** Correct (dilation property: center, pre - image, image vertices are collinear).
- **"The value of the scale factor, 6, is greater than 1."** Correct (enlargement needs \( k>1 \), 6>1).
- **"The triangle image is closer to the point of dilation than the pre - image."** Wrong (image is farther).

So the correct statements are:

1. The vertices of the pre - image are collinear with the vertices of the image.
2. The value of the scale factor, 6, is greater than 1. (or "The triangle pre - image is closer to the point of dilation than the image" – but let's confirm with dilation properties)

Wait, let's take the scale factor calculation. Let the distance from \( P \) to \( T \) be \( 2 \), to \( T' \) be \( 10 \). So scale factor \( k=\frac{PT'}{PT}=\frac{10}{2} = 5 \), which is \( k>1 \) (enlargement). So:

- Collinearity: Pre - image and image vertices lie on lines through \( P \), so they are collinear with \( P \) and each other. So "The vertices of the pre - image are collinear with the vertices of the image" is correct (since \( P \), \( T \), \( T' \) are collinear; same for \( S \), \( S' \) and \( U \), \( U' \)).

- "The value of the scale factor, 6, is greater than 1": Since \( k = 5>1 \), and 6>1, the statement is correct in terms of the scale factor being greater than 1 (the number 6 is just an example of a number greater than 1, which is the case for enlargement).

- "The triangle pre - image is closer to the point of dilation than the image": Since \( PT = 2 \), \( PT'=10 \), pre - image vertex \( T \) is at distance 2 from \( P \), image vertex \( T' \) at 10. So pre - image is closer. So this is also correct. Wait, now I'm confused. Let's re - check the problem statement: "Which statements justify that the dilation of triangle STU is an enlargement (increasing its size) by the magnitude of the scale factor? Choose two correct answers."

Enlargement means \( k>1 \). So:

- Statement 3: "The vertices of the pre - image are collinear with the vertices of the image." This is a property of dilation (collinearity with center), but does it justify enlargement? No, collinearity is true for any dilation (reduction or enlargement).

- Statement 4: "The value of the scale factor, 6, is greater than 1." Since enlargement requires \( k>1 \), this justifies that it's an enlargement (scale factor magnitude > 1).

- Statement 1: "The triangle pre - image is closer to the point of dilation than the image." For \( k>1 \), image is farther from center, so pre - image is closer. This also justifies enlargement (since if pre - image were closer, image is larger, so enlargement).

- Statement 2: "The value of the scale factor, \( \frac{1}{6} \), is between 0 and 1." This is reduction, so wrong.

- Statement 5: "The triangle image is closer to the point of dilation than the pre - image." Wrong, as image is farther for enlargement.

So the two correct statements are:

- The vertices of the pre - image are collinear with the vertices of the image. (Wait, no, collinearity is for any dilation. Wait, maybe I was wrong. Let's think again. The key to enlargement is \( k>1 \), so statements that show \( k>1 \) or properties of enlargement.

Wait, the scale factor is \( k=\frac{\text{image length}}{\text{pre - image length}} \). If \( k>1 \), it's enlargement.

So:

- "The value of the scale factor, 6, is greater than 1": 6>1, so this shows \( k>1 \) (enlargement).

- "The triangle pre - image is closer to the point of dilation than the image": For \( k>1 \), image is farther, so pre - image is closer. This is a property of enlargement (since larger image is farther from center).

- "The vertices of the pre - image are collinear with the vertices of the image": This is true for any dilation, so it doesn't justify enlargement (could be reduction too).

So the two correct statements are:

1. The value of the scale factor, 6, is greater than 1.

2. The triangle pre - image is closer to the point of dilation than the image.

(Or "The vertices of the pre - image are collinear with the vertices of the image" – but collinearity is for all dilations. So the better justifications are the ones related to \( k>1 \) and distance from center.)

Wait, let's check the distance. If the pre - image is closer to the dilation center than the image, that means the image is farther, so it's larger (enlargement). And a scale factor greater than 1 also means enlargement. So these two statements are correct.

So the correct options (assuming the options are labeled as per the image, but since we can't see the labels, we'll list the statements):

- The vertices of the pre - image are collinear with the vertices of the image. (Wait, no, collinearity is a property of dilation, not specific to enlargement. So maybe the two correct are:

1. The value of the scale factor, 6, is greater than 1.

2. The triangle pre - image is closer to the point of dilation than the image.

(Or "The vertices of the pre - image are collinear with the vertices of the image" – but I think the intended answers are the ones with scale factor >1 and pre - image closer.)

Answer:

The two correct statements are:

• The vertices of the pre - image are collinear with the vertices of the image.
• The value of the scale factor, 6, is greater than 1.

(Or, depending on the diagram's actual scale, but based on dilation properties for enlargement, these two (or the pre - image closer and scale factor >1) are correct. If we take the scale factor calculation from \( PT = 2 \), \( PT'=10 \), scale factor \( k = 5>1 \), so "The value of the scale factor, 6, is greater than 1" is correct as 6>1 (enlargement scale factor is >1), and "The triangle pre - image is closer to the point of dilation than the image" is correct as image is farther for \( k>1 \).)