How to I find the length of a side on a triangle?

Question

How do I find line AB in this if ac is $6cm$, and bc is $14cm$? angle A is $59^\circ$, B is $55^\circ$, and C is $66^\circ$. (not to scale)

thanks in advance

Answer 1

Without further information, you cannot say. With no knowledge about the angles, there are infinitely many triangles that have the lengths of two legs as $6$ and $14$.

Answer 2

If $AB=x$ cm

we need

$x+14>6,$ which is always true as $x>0$

$ 14+6>x \implies 20>x$

$6+x>14\implies x>8$

So, $x$ can assume any values between $(8,20)$

Answer 3

Hint: with the added angles given, use the Law of Sines to compute the length of $\overline{AB}$. Taking angle B to be $55^\circ$

For angles A, B, C of a triangle, with $a$ being the length of the side opposite angle A, and so on, we have the following equality of ratios: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac {c}{\sin C}\tag{Law of Sines}$$

$$\frac {14}{\sin(59^\circ)} = \frac {|AB|}{\sin(66^\circ)} = \frac{6}{\sin(55^\circ)}$$

As you can see, the first and last fraction are not equal, so no such triangle can exist.

Answer 4

From $\frac {AB}{\sin C}=\frac{BC}{\sin A}$ we get $AB=13.14$cm. From $\frac {AB}{\sin C}=\frac{AC}{\sin B}$ we get $AB=6.28$cm. Note that not only the results differ significantly, the second one even dos not obey the triangle inequality. Where did you get your numbers from?