# I cut a triangle ABC out of a piece of paper. I found out that sinA+sinB+sinC=5/2 and a/sinA=16. What is the triangle’s perimeter?

Mar 20, 2018

Permeter $= \left(a + b + c\right) = 40$

#### Explanation:

By sine rule for a $\Delta A B C$ we have the relation

$\frac{a}{\sin} A = \frac{b}{\sin} B = \frac{c}{\sin} C \ldots . . \left[1\right]$,

where $a , b , c$ are opposite sides of the angles $A , B , C$ respectively

Given $\frac{a}{\sin} A = 16. \ldots . \left[2\right]$

and $\sin A + \sin B + \sin C = \frac{5}{2.} \ldots \ldots . . \left[3\right]$

combining [1] and [2] we get

$\frac{a}{\sin} A = \frac{b}{\sin} B = \frac{c}{\sin} C = 16$

Hence

$\sin A = \frac{a}{16}$

$\sin B = \frac{b}{16}$

$\sin C = \frac{c}{16}$

Inserting these in [3] we get

$\frac{a}{16} + \frac{b}{16} + \frac{c}{16} = \frac{5}{2}$

$\implies \left(a + b + c\right) = \frac{5}{2} \times 16 = 40$

So Perimeter $= \left(a + b + c\right) = 40$