# A triangle has sides A, B, and C. Sides A and B are of lengths 9 and 8, respectively, and the angle between A and B is pi/4. What is the length of side C?

Jan 7, 2016

C = $\sqrt{82}$

#### Explanation:

Note that $\frac{\pi}{4} \text{ radians } \to {45}^{o}$

Let $\angle c \mathrm{db} = \frac{\pi}{2} \text{ that is: } {90}^{o}$

As$\angle \mathrm{dc} b = \frac{\pi}{4} \text{ then } \angle c b d = \frac{\pi}{4}$

Thus it follows that $b d = \mathrm{dc} = 9$

Thus $a d = 9 - 8 = 1$

By Pythagoras: ${\left(b d\right)}^{2} + {\left(\mathrm{da}\right)}^{2} = {C}^{2}$

$\implies C = \sqrt{{\left(b d\right)}^{2} + {\left(\mathrm{da}\right)}^{2}}$

$\implies C = \sqrt{{9}^{2} + {1}^{2}} = \sqrt{82}$

Whilst 82 is not prime it is a product of prime numbers. As far as I am aware the only way of showing the precise value of C is to express it in this form.

Thus C = $\sqrt{82}$