A triangle has sides A, B, and C. The angle between sides A and B is $\frac{5 π}{6}$ $(5pi)/6$ . If side C has a length of $2$ $2$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what are the lengths of sides A and B?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{5 π}{6}$ $(5pi)/6$ . If side C has a length of $2$ $2$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what are the lengths of sides A and B?

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Answer 1 · 2017-10-24T13:29:47

A = B = 1.035

Explanation:

This one is deceptively easy, I think.

You know that the interior angles of a triangle add up to $π$ $pi$ radians (or 180 degrees). You are given 2 of the angles, so you can calculate the angle between sides A and C = $π - (\frac{π}{12} + \frac{5 π}{6})$ $pi - (pi/12 + (5pi)/6)$

...this comes out to $\frac{π}{12}$ $pi/12$ . This is the same as the angle between sides B & C.

So you know you have an isosceles triangle, and therefore sides A and B are equal.

From your equilateral triangle, you can create two congruent right triangles, with hypotenuse A (or B), and base of length $C_{1}$ $C_1$ and $C_{2}$ $C_2$ . ( $C_{1} + C_{2} = 2$ $C_1 + C_2 = 2$ , and since the two right triangles are equal, $C_{1} = C_{2}$ $C_1 = C_2$ , so $C_{1} = C_{2} = 1$ $C_1 = C_2 = 1$ )

So now, using just a little trig, we know: $\frac{C_{1}}{A} = cos (\frac{π}{12})$ $C_1/A = cos(pi/12)$
...since $C_{1} = 1$ $C_1 = 1$ , we have $\frac{1}{A} = cos (\frac{π}{12})$ $1/A = cos(pi/12)$
Therefore, $A = \frac{1}{cos (\frac{π}{12})}$ $A = 1/cos(pi/12)$

$A = 1.035$ $A = 1.035$ (rounding)

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{5 π}{6}$ $(5pi)/6$ . If side C has a length of $2$ $2$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what are the lengths of sides A and B?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{5 π}{6}$ $(5pi)/6$ . If side C has a length of $2$ $2$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what are the lengths of sides A and B?