Question
Sasha says that she drew an acute isosceles triangle with side lengths of 6 cm 9 cm and 12 cm and angles of 30 degrees 50 degrees and 100 degrees Danielle says that is not possible Explain using sides and angles who is correct
Answers
GPT-4o mini
To determine who is correct, let's analyze the triangle that Sasha claims to have drawn with the given side lengths (6 cm, 9 cm, and 12 cm) and angle measures (30 degrees, 50 degrees, and 100 degrees).
### Step 1: Check the triangle inequality
First, we need to ensure that the sides can form a triangle based on the triangle inequality, which states that the sum of the lengths of any two sides must be greater than the length of the third side.
1. **6 cm + 9 cm > 12 cm:**
\(15 > 12\) (True)
2. **6 cm + 12 cm > 9 cm:**
\(18 > 9\) (True)
3. **9 cm + 12 cm > 6 cm:**
\(21 > 6\) (True)
Since all conditions of the triangle inequality are satisfied, it is possible to form a triangle with these side lengths.
### Step 2: Determine the angles using the side lengths
Next, we need to find out if the triangle is acute and isosceles with those specified angles. An isosceles triangle has at least two equal sides and consequently at least two equal angles.
#### Determine the longest side:
The longest side is 12 cm. The corresponding angle in an isosceles triangle should be opposite one of the equal sides, which must be less than 90 degrees in an acute triangle.
Given the three angles specified by Sasha (30°, 50°, and 100°):
1. **100°** is an obtuse angle (greater than 90°) and indicates that this triangle cannot be acute.
2. For an isosceles triangle to exist, it must have two equal sides and, consequently, two equal angles. However, none of the angles given (30°, 50°, and 100°) are equal.
#### Check angle sums:
The sum of angles in any triangle must equal 180°.
\(30° + 50° + 100° = 180°\) (True)
### Conclusion:
Although the side lengths satisfy the triangle inequality, the angles do not form an acute triangle as Sasha claims. The presence of a 100° angle violates the requirement for the triangle to be acute (which must have all angles less than 90°).
**Thus, Danielle is correct**: it is not possible for Sasha to have drawn an acute isosceles triangle with the provided side lengths and angles. The triangle described would actually be obtuse and not isosceles.
### Step 1: Check the triangle inequality
First, we need to ensure that the sides can form a triangle based on the triangle inequality, which states that the sum of the lengths of any two sides must be greater than the length of the third side.
1. **6 cm + 9 cm > 12 cm:**
\(15 > 12\) (True)
2. **6 cm + 12 cm > 9 cm:**
\(18 > 9\) (True)
3. **9 cm + 12 cm > 6 cm:**
\(21 > 6\) (True)
Since all conditions of the triangle inequality are satisfied, it is possible to form a triangle with these side lengths.
### Step 2: Determine the angles using the side lengths
Next, we need to find out if the triangle is acute and isosceles with those specified angles. An isosceles triangle has at least two equal sides and consequently at least two equal angles.
#### Determine the longest side:
The longest side is 12 cm. The corresponding angle in an isosceles triangle should be opposite one of the equal sides, which must be less than 90 degrees in an acute triangle.
Given the three angles specified by Sasha (30°, 50°, and 100°):
1. **100°** is an obtuse angle (greater than 90°) and indicates that this triangle cannot be acute.
2. For an isosceles triangle to exist, it must have two equal sides and, consequently, two equal angles. However, none of the angles given (30°, 50°, and 100°) are equal.
#### Check angle sums:
The sum of angles in any triangle must equal 180°.
\(30° + 50° + 100° = 180°\) (True)
### Conclusion:
Although the side lengths satisfy the triangle inequality, the angles do not form an acute triangle as Sasha claims. The presence of a 100° angle violates the requirement for the triangle to be acute (which must have all angles less than 90°).
**Thus, Danielle is correct**: it is not possible for Sasha to have drawn an acute isosceles triangle with the provided side lengths and angles. The triangle described would actually be obtuse and not isosceles.