Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To determine which angle in a triangle has the greatest measure given the side lengths [tex]\( BC = 9 \)[/tex], [tex]\( AB = 7 \)[/tex], and [tex]\( AC = 13 \)[/tex], we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], and opposite angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] respectively:
[tex]\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \][/tex]
[tex]\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} \][/tex]
[tex]\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]
Given the side lengths:
- [tex]\( a = AC = 13 \)[/tex]
- [tex]\( b = AB = 7 \)[/tex]
- [tex]\( c = BC = 9 \)[/tex]
We can calculate the cosine of each angle:
1. To find [tex]\(\cos(A)\)[/tex]:
[tex]\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} = \frac{7^2 + 9^2 - 13^2}{2 \times 7 \times 9} \][/tex]
[tex]\[ \cos(A) = \frac{49 + 81 - 169}{126} = \frac{-39}{126} = -0.3095 \][/tex]
2. To find [tex]\(\cos(B)\)[/tex]:
[tex]\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} = \frac{13^2 + 9^2 - 7^2}{2 \times 13 \times 9} \][/tex]
[tex]\[ \cos(B) = \frac{169 + 81 - 49}{234} = \frac{201}{234} = 0.858 \][/tex]
3. To find [tex]\(\cos(C)\)[/tex]:
[tex]\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} = \frac{13^2 + 7^2 - 9^2}{2 \times 13 \times 7} \][/tex]
[tex]\[ \cos(C) = \frac{169 + 49 - 81}{182} = \frac{137}{182} = 0.753 \][/tex]
Next, we convert these cosine values to angles in degrees:
- [tex]\(\angle A \approx 108.03^\circ\)[/tex]
- [tex]\(\angle B \approx 41.17^\circ\)[/tex]
- [tex]\(\angle C \approx 30.80^\circ\)[/tex]
Now we compare the angles:
- [tex]\(\angle A = 108.03^\circ\)[/tex]
- [tex]\(\angle B = 41.17^\circ\)[/tex]
- [tex]\(\angle C = 30.80^\circ\)[/tex]
The greatest angle is [tex]\(\angle A = 108.03^\circ\)[/tex].
Thus, the angle in the triangle with the greatest measure is:
[tex]\[ \boxed{\angle A} \][/tex]
[tex]\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \][/tex]
[tex]\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} \][/tex]
[tex]\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]
Given the side lengths:
- [tex]\( a = AC = 13 \)[/tex]
- [tex]\( b = AB = 7 \)[/tex]
- [tex]\( c = BC = 9 \)[/tex]
We can calculate the cosine of each angle:
1. To find [tex]\(\cos(A)\)[/tex]:
[tex]\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} = \frac{7^2 + 9^2 - 13^2}{2 \times 7 \times 9} \][/tex]
[tex]\[ \cos(A) = \frac{49 + 81 - 169}{126} = \frac{-39}{126} = -0.3095 \][/tex]
2. To find [tex]\(\cos(B)\)[/tex]:
[tex]\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} = \frac{13^2 + 9^2 - 7^2}{2 \times 13 \times 9} \][/tex]
[tex]\[ \cos(B) = \frac{169 + 81 - 49}{234} = \frac{201}{234} = 0.858 \][/tex]
3. To find [tex]\(\cos(C)\)[/tex]:
[tex]\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} = \frac{13^2 + 7^2 - 9^2}{2 \times 13 \times 7} \][/tex]
[tex]\[ \cos(C) = \frac{169 + 49 - 81}{182} = \frac{137}{182} = 0.753 \][/tex]
Next, we convert these cosine values to angles in degrees:
- [tex]\(\angle A \approx 108.03^\circ\)[/tex]
- [tex]\(\angle B \approx 41.17^\circ\)[/tex]
- [tex]\(\angle C \approx 30.80^\circ\)[/tex]
Now we compare the angles:
- [tex]\(\angle A = 108.03^\circ\)[/tex]
- [tex]\(\angle B = 41.17^\circ\)[/tex]
- [tex]\(\angle C = 30.80^\circ\)[/tex]
The greatest angle is [tex]\(\angle A = 108.03^\circ\)[/tex].
Thus, the angle in the triangle with the greatest measure is:
[tex]\[ \boxed{\angle A} \][/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.