Question 1-8
A triangle is graphed on a coordinate grid with representing the location of a vertex on the triangle. The triangle is transformed using the rule ( ,
.
Which statement is true?
If is greater than 1, the image of the triangle is smaller than the pre-image of the triangle.
If is between 0 and 1, the image of the triangle is smaller than the pre-image of the triangle.
If is between 0 and 1 , the image of the triangle is larger than the pre-image of the triangle.

Ask by Sullivan Hanson. in the United States

Jan 27,2025

Question

Question 1-8
A triangle is graphed on a coordinate grid with representing the location of a vertex on the triangle. The triangle is transformed using the rule ( ,
.
Which statement is true?
If is greater than 1, the image of the triangle is smaller than the pre-image of the triangle.
If is between 0 and 1, the image of the triangle is smaller than the pre-image of the triangle.
If is between 0 and 1 , the image of the triangle is larger than the pre-image of the triangle.

Ask by Sullivan Hanson. in the United States

Jan 27,2025

UpStudy AI · Accepted Answer

When a triangle is transformed using the rule $$(x, y) \rightarrow (a x, a y)$$, the value of $$a$$ determines how the size of the triangle changes:

1. **If $$a > 1$$:**  
   The transformation scales the triangle by a factor greater than 1, making the image **larger** than the pre-image.  
   *Example:* If $$a = 2$$, every coordinate of the triangle is doubled, effectively doubling the size of the triangle.

2. **If $$0 < a < 1$$:**  
   The transformation scales the triangle by a factor between 0 and 1, making the image **smaller** than the pre-image.  
   *Example:* If $$a = 0.5$$, every coordinate of the triangle is halved, effectively reducing the size of the triangle by half.

3. **If $$a = 1$$:**  
   The triangle remains unchanged because each coordinate is multiplied by 1.

Given the options:

- **First Statement:** *"If $$a$$ is greater than 1, the image of the triangle is smaller than the pre-image of the triangle."*  
  **This statement is **false** because the image becomes larger, not smaller.**

- **Second Statement:** *"If $$a$$ is between 0 and 1, the image of the triangle is smaller than the pre-image of the triangle."*  
  **This statement is **true** as explained above.**

- **Third Statement:** *"If $$a$$ is between 0 and 1, the image of the triangle is larger than the pre-image of the triangle."*  
  **This statement is **false** because the image becomes smaller, not larger.**

**Therefore, the correct statement is:**

**If $$a$$ is between 0 and 1, the image of the triangle is smaller than the pre-image of the triangle.**
If $$a$$ is between 0 and 1, the image of the triangle is smaller than the pre-image.

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