Question
Makayla wants to classify the triangle below. Select all of the terms that apply. 7.1 cm 5 cm 5 cm A. acute B. isosceles C. obtuse D. right of 14 Answered
Answers
GPT-4o mini
To classify the triangle with side lengths of 7.1 cm, 5 cm, and 5 cm, we can consider the following characteristics:
1. **Isosceles:** A triangle is classified as isosceles if it has at least two sides that are equal in length. In this case, the sides measuring 5 cm are equal, so this triangle is isosceles.
2. **Acute:** A triangle is acute if all its angles are less than 90 degrees. We can check this by examining the angles. However, to determine the type of triangle based on the side lengths, we can apply the triangle inequality theorem. The longest side (7.1 cm) must be less than the sum of the other two sides (5 cm + 5 cm = 10 cm), which it is. To check if it is acute, we compare the squares of the lengths:
- \( a^2 + b^2 > c^2 \)
- Where \( a \) and \( b \) are the equal sides (5 cm) and \( c \) is the longest side (7.1 cm).
- \( 5^2 + 5^2 = 25 + 25 = 50 \)
- \( 7.1^2 = 50.41 \)
Since \( 50 < 50.41 \), the triangle has at least one angle that is obtuse, meaning it is not acute.
3. **Obtuse:** A triangle is obtuse if one of its angles is greater than 90 degrees. Based on the calculations above, the triangle is indeed obtuse.
4. **Right:** A triangle is right if one of its angles is exactly 90 degrees. Based on our findings, it cannot be classified as right since there is at least one obtuse angle.
Given this information, the applicable classifications for the triangle are:
- B. isosceles
- C. obtuse
Therefore, the answers are **B and C**.
1. **Isosceles:** A triangle is classified as isosceles if it has at least two sides that are equal in length. In this case, the sides measuring 5 cm are equal, so this triangle is isosceles.
2. **Acute:** A triangle is acute if all its angles are less than 90 degrees. We can check this by examining the angles. However, to determine the type of triangle based on the side lengths, we can apply the triangle inequality theorem. The longest side (7.1 cm) must be less than the sum of the other two sides (5 cm + 5 cm = 10 cm), which it is. To check if it is acute, we compare the squares of the lengths:
- \( a^2 + b^2 > c^2 \)
- Where \( a \) and \( b \) are the equal sides (5 cm) and \( c \) is the longest side (7.1 cm).
- \( 5^2 + 5^2 = 25 + 25 = 50 \)
- \( 7.1^2 = 50.41 \)
Since \( 50 < 50.41 \), the triangle has at least one angle that is obtuse, meaning it is not acute.
3. **Obtuse:** A triangle is obtuse if one of its angles is greater than 90 degrees. Based on the calculations above, the triangle is indeed obtuse.
4. **Right:** A triangle is right if one of its angles is exactly 90 degrees. Based on our findings, it cannot be classified as right since there is at least one obtuse angle.
Given this information, the applicable classifications for the triangle are:
- B. isosceles
- C. obtuse
Therefore, the answers are **B and C**.