Question

How do you find the perimeter of an isosceles triangle whose base is eight and vertex angle is twenty degree?

Answer 1

Hint: The given question is of isosceles triangle and to find the perimeter of the triangle here we should know the formulae for the parameter of the triangle which is sum of all sides, and to obtain this parameter we should first find the sides length of this triangle.
Formulae Used: Parameter of triangle= sum of all sides.

Complete step by step solution:
Here we know the length of one side and the vertex angle, in order to obtain the length of the other sides; we have to calculate it by using basic trigonometry, on solving we get:

Here in the given figure The line AD is the median for the base, and we know that in the isosceles triangle median drawn to a side will bisect the vertex angle from which the median is extended, and accordingly in the figure the angles are shown.
Now here will use the “cos” function of trigonometry in order to get the length of the sides of the triangles.
We know:
 \[\cos \theta = \dfrac{{base}}{{hypo\tan eous}}\]
Here we know that in isosceles triangle two sides are of same length and the angles associated with them are also same, on solving we get:
 \[
   \Rightarrow In\,\Delta ACD \\
   \Rightarrow \cos 80 = \dfrac{4}{{AC}} \\
   \Rightarrow AC = \dfrac{4}{{\cos 80}} = 23.035 \;
 \]
Hence parameter of triangle is:
 \[ \Rightarrow AB + BC + CA = 23.035 + 8 + 23.035 = 54.07\]
So, the correct answer is “54.07 Units”.

Note: Perimeter means the sum of the distance of the boundaries covered by any two dimensional shape, or any boundary, here for this question parameter of the triangle was needed to be obtained so we first find the length of all sides and then calculate the parameter.