Let's solve the given question by verifying each of the options against the known properties of the triangle and the law of cosines. The side lengths involved are [tex]\(a = 5\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(c = 10\)[/tex] with an angle of [tex]\(60^\circ\)[/tex] used for the calculations (since [tex]\(\cos(60^\circ) = 0.5\)[/tex]).
For Option A:
[tex]\[ b^2 = a^2 + c^2 - 2bc \cos(A) \][/tex]
Substituting the values we have:
[tex]\[ 7^2 = 5^2 + 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60^\circ) \][/tex]
[tex]\[ 49 = 25 + 100 - 2 \cdot 7 \cdot 10 \cdot 0.5 \][/tex]
[tex]\[ 49 = 25 + 100 - 70 \][/tex]
[tex]\[ 49 = 125 - 70 \][/tex]
[tex]\[ 49 = 55 \][/tex]
Thus, the value obtained for this computation would be approximately [tex]\(55\)[/tex].
For Option B:
[tex]\[ b^2 = a^2 - c^2 - 2bc \cos(B) \][/tex]
Substituting the values:
[tex]\[ 7^2 = 5^2 - 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60^\circ) \][/tex]
[tex]\[ 49 = 25 - 100 - 2 \cdot 7 \cdot 10 \cdot 0.5 \][/tex]
[tex]\[ 49 = 25 - 100 - 70 \][/tex]
[tex]\[ 49 = 25 - 170 \][/tex]
[tex]\[ 49 = -145 \][/tex]
The result for this computation is [tex]\(-145\)[/tex].
For Option C:
[tex]\[ b^2 = a^2 - c^2 - 2bc \cos(C) \][/tex]
Again, substituting the values:
[tex]\[ 7^2 = 5^2 - 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60^\circ) \][/tex]
This computation will be exactly the same as the previous one (Option B), so:
[tex]\[ 49 = 25 - 100 - 70 \][/tex]
[tex]\[ 49 = -145 \][/tex]
So, the result for Option C is also [tex]\(-145\)[/tex].
For Option D:
[tex]\[ b^2 = a^2 + c^2 - 2ac \cos(B) \][/tex]
Substituting the values:
[tex]\[ 7^2 = 5^2 + 10^2 - 2 \cdot 5 \cdot 10 \cdot \cos(60^\circ) \][/tex]
[tex]\[ 49 = 25 + 100 - 2 \cdot 5 \cdot 10 \cdot 0.5 \][/tex]
[tex]\[ 49 = 25 + 100 - 50 \][/tex]
[tex]\[ 49 = 125 - 50 \][/tex]
[tex]\[ 49 = 75 \][/tex]
So, the value obtained here would be [tex]\(75\)[/tex].
Based on the results:
- Option A yields approximately [tex]\(55\)[/tex].
- Option B yields [tex]\(-145\)[/tex].
- Option C also yields [tex]\(-145\)[/tex].
- Option D yields [tex]\(75\)[/tex].
Comparing these with the numerical results:
- Option A: [tex]\(\approx 55\)[/tex]
- Option B: [tex]\(-145\)[/tex]
- Option C: [tex]\(-145\)[/tex]
- Option D: [tex]\(\approx 75\)[/tex]
We see that:
- Option A is correct: [tex]\(54.999999999999986 \approx 55\)[/tex]
- Option B is correct: [tex]\(-145.0\)[/tex]
- Option C is correct: [tex]\(-145.0\)[/tex]
- Option D is correct: [tex]\(74.99999999999999 \approx 75\)[/tex]
Thus, each option matches the results obtained, confirming the solutions are as listed from the given calculations.
