Calculating Triangle Third Side using Law of Cosines
We are given two sides of a triangle and the included angle. We need to find the length of the third side.
- Side 1 length, $a = 7$ cm.
- Side 2 length, $b = 8$ cm.
- Included angle, $C = 60^\circ$.
Applying the Law of Cosines
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is:
$c^2 = a^2 + b^2 - 2ab \cos(C)$
Where $c$ is the length of the third side (opposite angle C).
Calculating the Third Side Length
Substitute the given values into the formula:
$c^2 = 7^2 + 8^2 - 2(7)(8) \cos(60^\circ)$
We know that $\cos(60^\circ) = \frac{1}{2}$. Substitute this value:
$c^2 = 49 + 64 - 2(7)(8) \times \frac{1}{2}$
Perform the calculations:
$c^2 = 113 - (112 \times \frac{1}{2})$
$c^2 = 113 - 56$
$c^2 = 57$
To find the length $c$, take the square root of both sides:
$c = \sqrt{57}$
The length of the third side is $\sqrt{57}$ cm.
Final Answer Selection
The calculated length matches Option A.
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