To determine who is correct, we need to check whether the triangle with sides of lengths 6 cm, 9 cm, and 12 cm can exist, and whether the angles of 30°, 50°, and 100° can correspond to those side lengths in an acute isosceles triangle.
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Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
- \(6 + 9 = 15 > 12\) (true)
- \(6 + 12 = 18 > 9\) (true)
- \(9 + 12 = 21 > 6\) (true)
Since all conditions of the Triangle Inequality are met, a triangle with these side lengths can exist.
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Type of Triangle: An isosceles triangle has at least two equal sides. In this case, none of the sides are equal (6 cm, 9 cm, and 12 cm), so it cannot be isosceles.
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Angle-Side Relationship: In any triangle, the angles opposite to the longest side must be the largest. Here, the longest side is 12 cm, which would be opposite the largest angle (100°, in this case).
However, since the triangle is supposed to be acute isosceles:
- In an acute triangle, all angles must be less than 90°.
- The provided angle of 100° means the triangle is obtuse, not acute.
Thus, Sasha's claims about the triangle being an acute isosceles triangle with the given sides and angles are incorrect.
Conclusion: Daniel is correct; it is not possible to have an acute isosceles triangle with the given side lengths and angles.
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