Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

Select the correct answer.

A triangle has side lengths [tex]\( BC = 9 \)[/tex], [tex]\( AB = 7 \)[/tex], and [tex]\( AC = 13 \)[/tex]. Which angle in the triangle has the greatest measure?

A. [tex]\( \angle C \)[/tex]
B. [tex]\( \angle B \)[/tex]
C. Cannot be determined
D. [tex]\( \angle A \)[/tex]


Sagot :

To determine which angle in the triangle [tex]\( \triangle ABC \)[/tex] has the greatest measure, we can employ the Law of Cosines.

Given:
- [tex]\( BC = a = 9 \)[/tex]
- [tex]\( AB = c = 7 \)[/tex]
- [tex]\( AC = b = 13 \)[/tex]

The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
[tex]\[ b^2 = a^2 + c^2 - 2ac \cos(B) \][/tex]
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]

First, let's find the cosine of each angle.

### Find [tex]\( \cos(\angle C) \)[/tex]:
[tex]\[ \cos(\angle C) = \frac{a^2 + b^2 - c^2}{2ab}\][/tex]
Substitute in the given side lengths:
[tex]\[ \cos(\angle C) = \frac{9^2 + 13^2 - 7^2}{2 \cdot 9 \cdot 13} \][/tex]
[tex]\[ \cos(\angle C) = \frac{81 + 169 - 49}{234} \][/tex]
[tex]\[ \cos(\angle C) = \frac{201}{234} \][/tex]
[tex]\[ \cos(\angle C) = \frac{67}{78} \approx 0.85897 \][/tex]

### Find [tex]\( \cos(\angle B) \)[/tex]:
[tex]\[ \cos(\angle B) = \frac{a^2 + c^2 - b^2}{2ac} \][/tex]
Substitute in the given side lengths:
[tex]\[ \cos(\angle B) = \frac{9^2 + 7^2 - 13^2}{2 \cdot 9 \cdot 7} \][/tex]
[tex]\[ \cos(\angle B) = \frac{81 + 49 - 169}{126} \][/tex]
[tex]\[ \cos(\angle B) = \frac{-39}{126} \][/tex]
[tex]\[ \cos(\angle B) = -\frac{13}{42} \approx -0.30952 \][/tex]

### Find [tex]\( \cos(\angle A) \)[/tex]:
[tex]\[ \cos(\angle A) = \frac{b^2 + c^2 - a^2}{2bc} \][/tex]
Substitute in the given side lengths:
[tex]\[ \cos(\angle A) = \frac{13^2 + 7^2 - 9^2}{2 \cdot 13 \cdot 7} \][/tex]
[tex]\[ \cos(\angle A) = \frac{169 + 49 - 81}{182} \][/tex]
[tex]\[ \cos(\angle A) = \frac{137}{182} \][/tex]
[tex]\[ \cos(\angle A) = \frac{137}{182} \approx 0.75275 \][/tex]

Now, to find the angles, we need to take the inverse cosine of each value found:

### Compute the angles:
[tex]\[ \angle C = \cos^{-1}(0.85897) \approx 30.8^\circ \][/tex]
[tex]\[ \angle B = \cos^{-1}(-0.30952) \approx 108.03^\circ \][/tex]
[tex]\[ \angle A = \cos^{-1}(0.75275) \approx 41.17^\circ \][/tex]

By comparison, [tex]\( \angle B \)[/tex] has the greatest measure at approximately [tex]\( 108.03^\circ \)[/tex].

Therefore, the angle in the triangle with the greatest measure is
[tex]\[ \boxed{\angle B} \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.