The length of the side AC of the triangle ABC whose side AB is of 8.2 cm and side BC is of 13.5 cm length with angle A of 81° is 12.1 cm approx.
What is law of cosine?
Let there is triangle ABC such that |AB| = a units, |AC| = b units, and |BC| = c units and the internal angle A is of θ degrees, then we have:
[tex]a^2 + b^2 -2ab\cos(\theta) = c^2[/tex]
(c is opposite side to angle A)
When one angle and two sides of a triangle are known, and we want to know the length of the remaining third side, then we can use the law of cosines.
We're specified that:
- Length of AB = |AB| = c = 8.2 cm
- |BC| = a = 13.5 cm
- m∠A = 81°
Let the third side's length = |AC| = b cm (don't mix this notation with the notation of the formula said above. We just used notation such that its the smaller case version of the vertex opposite to that side, for example, opposite to AC lies b).
For using formula, we just need to take care that the angle θ is the angle opposite to the side which is going to be on one side (the notation c^2 in the formula given and here since we know the angle A, so the side opposite to A which is BC will be used in one side of the cosine rule, as shown below).
The side opposite to the angle A is BC, thus, we get:
[tex]|BC|^2 = |AB|^2 + |AC|^2 -2|AB||AC| \cos(m\angle A)\\\\13.5^2 = 8.2^2 + b^2 -2(8.2)b \cos(81^\circ)\\\\b^2 - 2.566 b - 115.01 \approx 0\\\\b \approx \dfrac{2.566 \pm \sqrt{2.566^2 - 4(-115.01)}}{2}\\\\b= 12.0835, -9.5175[/tex]
b represents side length, therefore positive. Thus, we obtained the length of the side |AC| approximately 12.1 cm
Thus, the length of the side AC of the triangle ABC whose side AB is of 8.2 cm and side BC is of 13.5 cm length with angle A of 81° is 12.1 cm approx.
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