To determine whether it is possible to draw an acute isosceles triangle with the given side lengths of 6 cm, 9 cm, and 12 cm, and angles of 30°, 50°, and 100°, we need to consider both the properties of triangles and the classifications of angles and sides.
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Triangle Inequality Theorem: For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Let's check the combinations of the provided side lengths:
- \( 6 + 9 = 15 \) is greater than \( 12 \).
- \( 6 + 12 = 18 \) is greater than \( 9 \).
- \( 9 + 12 = 21 \) is greater than \( 6 \).
Since all combinations satisfy the triangle inequality theorem, it is possible to form a triangle with these side lengths.
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Angle Considerations: A triangle can only be classified as acute if each of its angles is less than 90°. The angles provided are 30°, 50°, and 100°. Since one angle (100°) is greater than 90°, this triangle cannot be acute.
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Isosceles Triangle Definition: An isosceles triangle has at least two equal sides. The side lengths provided are 6 cm, 9 cm, and 12 cm, and no two sides are equal, meaning we do not form an isosceles triangle.
Conclusion:
Based on the analysis:
- No, it is not possible to draw an acute isosceles triangle with the side lengths of 6 cm, 9 cm, and 12 cm and angles of 30°, 50°, and 100°. The triangle cannot be acute due to the presence of a 100° angle, and it cannot be isosceles because no two sides are of equal length.
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