To determine whether Sasha's statement about the isosceles triangle is correct, we need to examine the properties of triangles, specifically regarding sides and angles.
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Definition of Isosceles Triangle: An isosceles triangle has at least two sides that are equal in length. In Sasha's case, the side lengths are 6 cm, 9 cm, and 12 cm. None of these sides are equal, which means the triangle cannot be isosceles.
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Angles of a Triangle: The sum of the angles in any triangle must be 180°. However, Sasha states that the triangle has angles of 30°, 50°, and 100°: \[ 30° + 50° + 100° = 180° \] The angles do sum up to 180°, which is valid for a triangle.
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Identifying the Type of Triangle: To be classified as an acute triangle, all its angles must be less than 90°. In Sasha's case, one of the angles is 100°, which is greater than 90°. Therefore, the triangle cannot be acute.
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Verifying the Triangle’s Properties with Sides: For a triangle to exist with the given side lengths of 6 cm, 9 cm, and 12 cm, the triangle inequality theorem must be satisfied, which states that the sum of the lengths of any two sides 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)
This shows that the side lengths can indeed form a triangle.
Conclusion: Sasha claimed that the triangle is an acute isosceles triangle with the specified side lengths and angles, but she is incorrect because it does not have two equal sides (thus is not isosceles) and one of the angles is obtuse (100°, thus it is not acute). Therefore, Danielle is correct in stating that Sasha’s description is not possible.