Question
Triangle △ABC, shown here, is dilated by a scale factor of 2 to form △A'B'C'.

Which statement is true?
A.
△A'B'C' has sides measuring 6 mm, 22 mm, and 24 mm.
B.
△A'B'C' has angles measuring 30°, 130°, and 200°.
C.
△A'B'C' has sides measuring 8 mm, 24 mm, and 26 mm.
D.
△A'B'C' has angles measuring 15°, 65°, and 100°.

Which statement is true?
A.
△A'B'C' has sides measuring 6 mm, 22 mm, and 24 mm.
B.
△A'B'C' has angles measuring 30°, 130°, and 200°.
C.
△A'B'C' has sides measuring 8 mm, 24 mm, and 26 mm.
D.
△A'B'C' has angles measuring 15°, 65°, and 100°.
Answers
Answered by
GPT-4o mini
When a triangle is dilated by a scale factor, the lengths of its sides increase by that scale factor, while the angles remain the same.
Given that \( \triangle ABC \) is dilated by a scale factor of 2 to form \( \triangle A'B'C' \), we can analyze the options:
1. **Option A:** States that \( \triangle A'B'C' \) has sides measuring 6 mm, 22 mm, and 24 mm.
- If the original side lengths were \( 3 \) mm, \( 11 \) mm, and \( 12 \) mm, then dilating by a factor of 2 would give us \( 6 \) mm, \( 22 \) mm, and \( 24 \) mm. This is possible, but without original measurements, we cannot be certain.
2. **Option B:** States that \( \triangle A'B'C' \) has angles measuring 30°, 130°, and 200°.
- The sum of the angles in a triangle must equal 180°. This option is impossible.
3. **Option C:** States that \( \triangle A'B'C' \) has sides measuring 8 mm, 24 mm, and 26 mm.
- If the original triangle's side lengths were \( 4 \) mm, \( 12 \) mm, and \( 13 \) mm, dilating by a factor of 2 would result in these side lengths. This is also possible.
4. **Option D:** States that \( \triangle A'B'C' \) has angles measuring 15°, 65°, and 100°.
- Again, the sum is 180°, which is valid, but we need to check against the dilation factors.
Since \( \triangle A'B'C' \) retains the same angles as \( \triangle ABC \), we should look for options that maintain angle relationships.
Options A and C are possible given the sides if the original measures are matched accordingly, while B is impossible, and D is feasible but we lack original angle measures.
Since both A and C are mathematically plausible, a definitive answer would depend on the original triangle's side lengths or angles. In absence of that specific information, we can select one answer.
If I had to select based on what is common in dilation with checking valid sides and angles:
C appears to be the most straightforward, as it exhibits reasonable logical possibility without creating impossible angles:
**C. \(\triangle A'B'C'\) has sides measuring 8 mm, 24 mm, and 26 mm.**
Given that \( \triangle ABC \) is dilated by a scale factor of 2 to form \( \triangle A'B'C' \), we can analyze the options:
1. **Option A:** States that \( \triangle A'B'C' \) has sides measuring 6 mm, 22 mm, and 24 mm.
- If the original side lengths were \( 3 \) mm, \( 11 \) mm, and \( 12 \) mm, then dilating by a factor of 2 would give us \( 6 \) mm, \( 22 \) mm, and \( 24 \) mm. This is possible, but without original measurements, we cannot be certain.
2. **Option B:** States that \( \triangle A'B'C' \) has angles measuring 30°, 130°, and 200°.
- The sum of the angles in a triangle must equal 180°. This option is impossible.
3. **Option C:** States that \( \triangle A'B'C' \) has sides measuring 8 mm, 24 mm, and 26 mm.
- If the original triangle's side lengths were \( 4 \) mm, \( 12 \) mm, and \( 13 \) mm, dilating by a factor of 2 would result in these side lengths. This is also possible.
4. **Option D:** States that \( \triangle A'B'C' \) has angles measuring 15°, 65°, and 100°.
- Again, the sum is 180°, which is valid, but we need to check against the dilation factors.
Since \( \triangle A'B'C' \) retains the same angles as \( \triangle ABC \), we should look for options that maintain angle relationships.
Options A and C are possible given the sides if the original measures are matched accordingly, while B is impossible, and D is feasible but we lack original angle measures.
Since both A and C are mathematically plausible, a definitive answer would depend on the original triangle's side lengths or angles. In absence of that specific information, we can select one answer.
If I had to select based on what is common in dilation with checking valid sides and angles:
C appears to be the most straightforward, as it exhibits reasonable logical possibility without creating impossible angles:
**C. \(\triangle A'B'C'\) has sides measuring 8 mm, 24 mm, and 26 mm.**
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