Question
In ΔABC, BC = 9, CA = 17, and AB = 13. Which list has the angles of ΔABC in order from largest to smallest?
Answer
Multiple Choice Answers
m, angle, B, comma, m, angle, A, comma, m, angle, Cm∠B,m∠A,m∠C
m, angle, C, comma, m, angle, A, comma, m, angle, Bm∠C,m∠A,m∠B
m, angle, C, comma, m, angle, B, comma, m, angle, Am∠C,m∠B,m∠A
m, angle, A, comma, m, angle, B, comma, m, angle, Cm∠A,m∠B,m∠C
m, angle, B, comma, m, angle, C, comma, m, angle, Am∠B,m∠C,m∠A
m, angle, A, comma, m, angle, C, comma, m, angle, Bm∠A,m∠C,m∠B
Answer
Multiple Choice Answers
m, angle, B, comma, m, angle, A, comma, m, angle, Cm∠B,m∠A,m∠C
m, angle, C, comma, m, angle, A, comma, m, angle, Bm∠C,m∠A,m∠B
m, angle, C, comma, m, angle, B, comma, m, angle, Am∠C,m∠B,m∠A
m, angle, A, comma, m, angle, B, comma, m, angle, Cm∠A,m∠B,m∠C
m, angle, B, comma, m, angle, C, comma, m, angle, Am∠B,m∠C,m∠A
m, angle, A, comma, m, angle, C, comma, m, angle, Bm∠A,m∠C,m∠B
Answers
GPT-4o mini
In triangle \( \Delta ABC \), we can determine the order of the angles by using the Law of Cosines to find which angle is the largest. The sides of the triangle are given as:
- \( BC = 9 \)
- \( CA = 17 \)
- \( AB = 13 \)
Using the Law of Cosines, we can calculate each angle.
1. **Finding \( m\angle A \)** (opposite side BC):
\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{9^2 + 13^2 - 17^2}{2 \times 9 \times 13}
\]
\[
= \frac{81 + 169 - 289}{234} = \frac{-39}{234} = -\frac{1}{6}
\]
2. **Finding \( m\angle B \)** (opposite side CA):
\[
\cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{17^2 + 13^2 - 9^2}{2 \times 17 \times 13}
\]
\[
= \frac{289 + 169 - 81}{442} = \frac{377}{442}
\]
3. **Finding \( m\angle C \)** (opposite side AB):
\[
\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{17^2 + 9^2 - 13^2}{2 \times 17 \times 9}
\]
\[
= \frac{289 + 81 - 169}{306} = \frac{201}{306} = \frac{67}{102}
\]
Next, we will compare the angles based on the cosine values calculated. Recall that the angle corresponding to the largest side is the largest angle in a triangle.
- Side \( CA = 17 \) is the longest, so \( m\angle B \) (opposite the longest side) should be the largest angle.
- Side \( AB = 13 \) is the next longest, so \( m\angle C \) will be the next largest angle.
- Side \( BC = 9 \) is the shortest, making \( m\angle A \) the smallest angle.
Therefore, the angles in descending order from largest to smallest are:
\( m\angle B, m\angle C, m\angle A \)
**The correct answer is:**
m∠B, m∠C, m∠A
- \( BC = 9 \)
- \( CA = 17 \)
- \( AB = 13 \)
Using the Law of Cosines, we can calculate each angle.
1. **Finding \( m\angle A \)** (opposite side BC):
\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{9^2 + 13^2 - 17^2}{2 \times 9 \times 13}
\]
\[
= \frac{81 + 169 - 289}{234} = \frac{-39}{234} = -\frac{1}{6}
\]
2. **Finding \( m\angle B \)** (opposite side CA):
\[
\cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{17^2 + 13^2 - 9^2}{2 \times 17 \times 13}
\]
\[
= \frac{289 + 169 - 81}{442} = \frac{377}{442}
\]
3. **Finding \( m\angle C \)** (opposite side AB):
\[
\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{17^2 + 9^2 - 13^2}{2 \times 17 \times 9}
\]
\[
= \frac{289 + 81 - 169}{306} = \frac{201}{306} = \frac{67}{102}
\]
Next, we will compare the angles based on the cosine values calculated. Recall that the angle corresponding to the largest side is the largest angle in a triangle.
- Side \( CA = 17 \) is the longest, so \( m\angle B \) (opposite the longest side) should be the largest angle.
- Side \( AB = 13 \) is the next longest, so \( m\angle C \) will be the next largest angle.
- Side \( BC = 9 \) is the shortest, making \( m\angle A \) the smallest angle.
Therefore, the angles in descending order from largest to smallest are:
\( m\angle B, m\angle C, m\angle A \)
**The correct answer is:**
m∠B, m∠C, m∠A