Question
Quadrilateral ABCD is dilated to create quadrilateral PQRS.
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image.png
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Which statements are true? Select all that apply.
Each side length of quadrilateral ABCD is 12
the corresponding side length of quadrilateral PQRS.
Quadrilateral PQRS is congruent to quadrilateral ABCD.
Each side length of quadrilateral ABCD is twice the corresponding side length of quadrilateral PQRS.
Each angle measure of quadrilateral ABCD is 12
the corresponding angle measure of quadrilateral PQRS.
Quadrilateral PQRS is similar to quadrilateral ABCD.
Each angle measure of quadrilateral ABCD is twice the corresponding angle measure of quadrilateral PQRS.
-
image.png
-
Which statements are true? Select all that apply.
Each side length of quadrilateral ABCD is 12
the corresponding side length of quadrilateral PQRS.
Quadrilateral PQRS is congruent to quadrilateral ABCD.
Each side length of quadrilateral ABCD is twice the corresponding side length of quadrilateral PQRS.
Each angle measure of quadrilateral ABCD is 12
the corresponding angle measure of quadrilateral PQRS.
Quadrilateral PQRS is similar to quadrilateral ABCD.
Each angle measure of quadrilateral ABCD is twice the corresponding angle measure of quadrilateral PQRS.
Answers
GPT-4o mini
To determine the true statements about the dilation of quadrilateral ABCD to create quadrilateral PQRS, let’s consider how dilation affects side lengths and angle measures.
1. **Each side length of quadrilateral ABCD is 1/2 the corresponding side length of quadrilateral PQRS.**
- This statement is **true** if PQRS is a dilation of ABCD where the scale factor is 2 (i.e., if each side of ABCD is half the length of the corresponding side of PQRS).
2. **Quadrilateral PQRS is congruent to quadrilateral ABCD.**
- This statement is **false** if a dilation is applied because dilations change the size of a shape but maintain its shape. Thus, congruence does not hold.
3. **Each side length of quadrilateral ABCD is twice the corresponding side length of quadrilateral PQRS.**
- This statement is **true** if we are following the previous logic of a scale factor of 1/2, leading to the conclusion that each side of ABCD is 1/2 the corresponding side of PQRS.
4. **Each angle measure of quadrilateral ABCD is 1/2 the corresponding angle measure of quadrilateral PQRS.**
- This statement is **false**; dilations do not change angle measures. Therefore, angle measures in ABCD will be equal to the corresponding angle measures in PQRS.
5. **Quadrilateral PQRS is similar to quadrilateral ABCD.**
- This statement is **true**; dilations maintain shape, and thus the two quadrilaterals are similar.
6. **Each angle measure of quadrilateral ABCD is twice the corresponding angle measure of quadrilateral PQRS.**
- This statement is **false**; again, angle measures remain unchanged under dilation.
Based on this analysis, the statements that are true are:
- Each side length of quadrilateral ABCD is 1/2 the corresponding side length of quadrilateral PQRS.
- Each side length of quadrilateral ABCD is twice the corresponding side length of quadrilateral PQRS.
- Quadrilateral PQRS is similar to quadrilateral ABCD.
1. **Each side length of quadrilateral ABCD is 1/2 the corresponding side length of quadrilateral PQRS.**
- This statement is **true** if PQRS is a dilation of ABCD where the scale factor is 2 (i.e., if each side of ABCD is half the length of the corresponding side of PQRS).
2. **Quadrilateral PQRS is congruent to quadrilateral ABCD.**
- This statement is **false** if a dilation is applied because dilations change the size of a shape but maintain its shape. Thus, congruence does not hold.
3. **Each side length of quadrilateral ABCD is twice the corresponding side length of quadrilateral PQRS.**
- This statement is **true** if we are following the previous logic of a scale factor of 1/2, leading to the conclusion that each side of ABCD is 1/2 the corresponding side of PQRS.
4. **Each angle measure of quadrilateral ABCD is 1/2 the corresponding angle measure of quadrilateral PQRS.**
- This statement is **false**; dilations do not change angle measures. Therefore, angle measures in ABCD will be equal to the corresponding angle measures in PQRS.
5. **Quadrilateral PQRS is similar to quadrilateral ABCD.**
- This statement is **true**; dilations maintain shape, and thus the two quadrilaterals are similar.
6. **Each angle measure of quadrilateral ABCD is twice the corresponding angle measure of quadrilateral PQRS.**
- This statement is **false**; again, angle measures remain unchanged under dilation.
Based on this analysis, the statements that are true are:
- Each side length of quadrilateral ABCD is 1/2 the corresponding side length of quadrilateral PQRS.
- Each side length of quadrilateral ABCD is twice the corresponding side length of quadrilateral PQRS.
- Quadrilateral PQRS is similar to quadrilateral ABCD.