Question

1. a triangle is mapped onto another triangle using a dilation with a scale factor of 3 centered at the origin. which of the following is true about the image compared to the pre - image? the image is congruent to the original; the image is similar but not congruent to the original; the image is neither congruent nor similar; the image has the same side lengths as the original

Explanation:

Step1: Recall Dilation Properties

Dilation is a transformation that scales a figure by a scale factor. For a scale factor \( k
eq 1 \), the image and pre - image are similar (same shape, different size) but not congruent (congruent requires same shape and size). Here, scale factor \( k = 3
eq1 \).

Step2: Analyze Each Option

• "The image is congruent to the original": Congruent means same size and shape. Dilation with \( k = 3 \) changes size, so this is false.
• "The image is similar but not congruent to the original": Similarity means same shape (corresponding angles equal, corresponding sides proportional). Dilation with \( k = 3 \) makes sides proportional (\( \text{new side length}=3\times\text{original side length} \)) and angles remain equal, so similar but not congruent (since size is different). This is true.
• "The image is neither congruent nor similar": Dilation preserves shape (so similar), so this is false.
• "The image has the same side lengths as the original": Dilation with \( k = 3 \) multiplies side lengths by 3, so side lengths change. This is false.

Answer:

The image is similar but not congruent to the original (the option with this text)