Question

Two corners of a triangle have angles of `(3 pi ) / 8` and `pi / 8`. If one side of the triangle has a length of `5`, what is the longest possible perimeter of the triangle?

Answer 1

use sine rule

Explanation:

i suggest you to find a piece of paper and a pencil to comprehend this explanation easier.

find the value of the remaining angle:
`pi = 3/8pi + 1/8pi + ?`
`? = pi - 3/8pi - 1/8pi = 1/2 pi`

lets give them names
`A=3/8 pi`
`B=1/8pi`
`C=1/2pi`
the smallest angle will face the shortest side of the triangle,
which means B(the smallest angle) is facing the shortest side,
and the other two sides are longer,
which means AC is the shortest side,
so the two other sides can have their longest length.
let's say AC is 5 (the length you given)

using sine rule, we can know
the ratio of the sine of an angle and the side which the angle is facing are the same:

`sinA/(BC) = sinB/(AC) = sinC/(AB)`

known:
`sin(1/8pi)/(5) = sin(3/8pi)/(BC) = sin(1/2pi)/(AB)`

with this, you can find the length of the other two sides when the shortest one is 5

I'll leave the rest for you, keep on going~