Question

Sasha says that she drew in acute isosceles triangle with silence of 6 cm 9 cm and 12 cm and angles of 30 50 and 100 Daniel says that is not possible explain using sides and angles who is correct

Answer 1

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.

1. **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.

2. **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.

3. **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.