Question
Sasha says that she drew an acute isosceles triangle with side legnths of 6cm, 9 cm, and 12 cm and angels of 30 degrees , 50 degrees, and 100 degrees. Danielle says this is not possible. Explain using sides and angles so is correct
Answers
GPT-4o mini
To determine if Sasha's claims about the triangle are possible or not, we need to analyze both the side lengths and the angles she provided.
1. **Triangle Sides**: Sasha reports the side lengths as 6 cm, 9 cm, and 12 cm. For any triangle, the triangle inequality theorem must hold, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let's check the conditions:
- \(6 + 9 = 15\) (which is greater than 12)
- \(6 + 12 = 18\) (which is greater than 9)
- \(9 + 12 = 21\) (which is greater than 6)
Since all these conditions are satisfied, the side lengths can indeed form a triangle.
2. **Triangle Angles**: Sasha claims the angles are 30 degrees, 50 degrees, and 100 degrees. The sum of the interior angles of a triangle must equal 180 degrees:
\[
30 + 50 + 100 = 180 \text{ degrees.}
\]
This condition is also satisfied.
3. **Type of Triangle**: An isosceles triangle has at least two sides of equal length. However, the side lengths provided (6 cm, 9 cm, and 12 cm) are all different. This means it cannot be classified as isosceles.
4. **Acute Triangle**: An acute triangle requires all angles to be less than 90 degrees. In Sasha's claim, she listed an angle of 100 degrees, which is clearly greater than 90 degrees, thus not all angles are acute.
Based on these analyses, Danielle is correct. Sasha's claim of an acute isosceles triangle with the specified angles and side lengths is impossible because:
- The triangle cannot be isosceles as all sides are of different lengths.
- The triangle cannot be acute since one of the angles is 100 degrees, which is obtuse.
Thus, Sasha's triangle is not possible under the described conditions.
1. **Triangle Sides**: Sasha reports the side lengths as 6 cm, 9 cm, and 12 cm. For any triangle, the triangle inequality theorem must hold, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let's check the conditions:
- \(6 + 9 = 15\) (which is greater than 12)
- \(6 + 12 = 18\) (which is greater than 9)
- \(9 + 12 = 21\) (which is greater than 6)
Since all these conditions are satisfied, the side lengths can indeed form a triangle.
2. **Triangle Angles**: Sasha claims the angles are 30 degrees, 50 degrees, and 100 degrees. The sum of the interior angles of a triangle must equal 180 degrees:
\[
30 + 50 + 100 = 180 \text{ degrees.}
\]
This condition is also satisfied.
3. **Type of Triangle**: An isosceles triangle has at least two sides of equal length. However, the side lengths provided (6 cm, 9 cm, and 12 cm) are all different. This means it cannot be classified as isosceles.
4. **Acute Triangle**: An acute triangle requires all angles to be less than 90 degrees. In Sasha's claim, she listed an angle of 100 degrees, which is clearly greater than 90 degrees, thus not all angles are acute.
Based on these analyses, Danielle is correct. Sasha's claim of an acute isosceles triangle with the specified angles and side lengths is impossible because:
- The triangle cannot be isosceles as all sides are of different lengths.
- The triangle cannot be acute since one of the angles is 100 degrees, which is obtuse.
Thus, Sasha's triangle is not possible under the described conditions.
Can u make it sentences