The Law of sines / geometry: iso. triangle w/ base b = 46...

[keri__lynn]

An isosceles triangle has a base of 46 centimeters and a vertex angle of 44 degrees. ( FIND THE PERIMETER)

can you please work me all the way through this to get the answer iam so confused and dont understand it

 

[soroban]

Hello, keri__lynn!

Did you make a sketch?

An isosceles triangle has a base of 46 centimeters and a vertex angle of 44 degrees.
Find the perimeter.

              A
              *
             / \
            /44d\
           /     \
        c /       \ b
         /         \
        /           \
       /             \
    B * - - - - - - - * C
             46

\(\displaystyle \text{Since }\Delta ABC\text{ is isosceles: }\:\angle B = \angle C\:\text{ and }\:b \,=\,c.\)
. . \(\displaystyle \text{and we find that: }\:\angle B \:=\:68^o\)

\(\displaystyle \text{Law of Sines: }\;\frac{b}{\sin B} \:=\:\frac{a}{\sin A} \quad\Rightarrow\quad \frac{b}{\sin 68^o} \:=\:\frac{46}{\sin 44^o}\)

. . \(\displaystyle \text{Hence: }\:b \:=\:\frac{46\sin68^o}{\sin44^o} \:=\:61.39774474 \:\approx\: 61.4\text{ cm}\)


\(\displaystyle \text{Therefore: }\:\text{Perimeter} \:=\:61.4 + 61.4 + 46 \:=\:\boxed{168.8\text{ cm}}\)

 

[Denis]

Quicker is: 23 / SIN(22)

Rule : side = (base / 2) / SIN(vertexangle / 2)

If you want to see/know why, drop a perpendicular from vertex to base; 2 identical right triangles, see that?