Question
Elijah was trying to determine whether one triangle is a dilation of the other. Which statements about his solution are true? Check all that apply.
pick
One triangle has side lengths of 3, 6, 7.5. Another triangle has side lengths of 5, 10, 12.5.
Elijah’s work:
3 is not a factor of 5, so it must be a relationship with addition.
3 + 2 = 5. 6 + 2 not-equals 10. 7.5 + 2 not-equals 12.5.
Therefore, the triangles are not proportional and cannot form a dilation.
Elijah did not know that scale factors can be fractions or decimals greater than zero.
Elijah did not know that dilations are proportional and the relationships are multiplicative, not additive.
Elijah should have compared the ratios Five-thirds = StartFraction 10 Over 6 EndFraction = StartFraction 12.5 Over 7.5 EndFraction to determine if they are equivalent.
Elijah needs to know which triangle is the pre-image and which triangle is the image in order to determine if they are proportional.
Elijah needs labels on the points to determine which sides correspond when writing the ratios.
pick
One triangle has side lengths of 3, 6, 7.5. Another triangle has side lengths of 5, 10, 12.5.
Elijah’s work:
3 is not a factor of 5, so it must be a relationship with addition.
3 + 2 = 5. 6 + 2 not-equals 10. 7.5 + 2 not-equals 12.5.
Therefore, the triangles are not proportional and cannot form a dilation.
Elijah did not know that scale factors can be fractions or decimals greater than zero.
Elijah did not know that dilations are proportional and the relationships are multiplicative, not additive.
Elijah should have compared the ratios Five-thirds = StartFraction 10 Over 6 EndFraction = StartFraction 12.5 Over 7.5 EndFraction to determine if they are equivalent.
Elijah needs to know which triangle is the pre-image and which triangle is the image in order to determine if they are proportional.
Elijah needs labels on the points to determine which sides correspond when writing the ratios.
Answers
GPT-4o mini
Based on the provided information about Elijah's solution, here's an evaluation of each statement regarding their truthfulness:
1. **Elijah did not know that scale factors can be fractions or decimals greater than zero.**
**True** - This statement is true because Elijah did not consider that the side lengths of one triangle could be dilated by a scale factor that is a fraction or decimal.
2. **Elijah did not know that dilations are proportional and the relationships are multiplicative, not additive.**
**True** - This statement is accurate. Dilation involves proportional relationships that require multiplication, not addition, which Elijah incorrectly applied in his reasoning.
3. **Elijah should have compared the ratios \( \frac{5}{3} = \frac{10}{6} = \frac{12.5}{7.5} \) to determine if they are equivalent.**
**True** - This statement is correct as comparing the ratios of the corresponding sides would clearly show whether the triangles are dilations of one another.
4. **Elijah needs to know which triangle is the pre-image and which triangle is the image in order to determine if they are proportional.**
**False** - This statement is not necessary for determining proportionality. The key is to discover if the ratios of corresponding sides are equivalent, regardless of which triangle is designated as the pre-image or image.
5. **Elijah needs labels on the points to determine which sides correspond when writing the ratios.**
**False** - While labeling can help in understanding which sides correspond, it is not an absolute necessity to determine if the triangles are dilations. The focus should be on the ratios of the sides.
In summary, the true statements are:
- Elijah did not know that scale factors can be fractions or decimals greater than zero.
- Elijah did not know that dilations are proportional and the relationships are multiplicative, not additive.
- Elijah should have compared the ratios \( \frac{5}{3} = \frac{10}{6} = \frac{12.5}{7.5} \) to determine if they are equivalent.
1. **Elijah did not know that scale factors can be fractions or decimals greater than zero.**
**True** - This statement is true because Elijah did not consider that the side lengths of one triangle could be dilated by a scale factor that is a fraction or decimal.
2. **Elijah did not know that dilations are proportional and the relationships are multiplicative, not additive.**
**True** - This statement is accurate. Dilation involves proportional relationships that require multiplication, not addition, which Elijah incorrectly applied in his reasoning.
3. **Elijah should have compared the ratios \( \frac{5}{3} = \frac{10}{6} = \frac{12.5}{7.5} \) to determine if they are equivalent.**
**True** - This statement is correct as comparing the ratios of the corresponding sides would clearly show whether the triangles are dilations of one another.
4. **Elijah needs to know which triangle is the pre-image and which triangle is the image in order to determine if they are proportional.**
**False** - This statement is not necessary for determining proportionality. The key is to discover if the ratios of corresponding sides are equivalent, regardless of which triangle is designated as the pre-image or image.
5. **Elijah needs labels on the points to determine which sides correspond when writing the ratios.**
**False** - While labeling can help in understanding which sides correspond, it is not an absolute necessity to determine if the triangles are dilations. The focus should be on the ratios of the sides.
In summary, the true statements are:
- Elijah did not know that scale factors can be fractions or decimals greater than zero.
- Elijah did not know that dilations are proportional and the relationships are multiplicative, not additive.
- Elijah should have compared the ratios \( \frac{5}{3} = \frac{10}{6} = \frac{12.5}{7.5} \) to determine if they are equivalent.