Problem 2. Given an isosceles triangle $ABC$ with base $AB$, where the length of the base $a$ is half the length of the leg $b$.

a) Calculate the lengths of the base $a$ and the leg $b$, if the perimeter of the triangle is $150 \text{~mm}$.

b) Calculate the length of the side of an equilateral triangle whose perimeter is equal to the perimeter of the given triangle.
Solution. a) For the length of the base $a$ and the length of the leg $b$, it holds that $b=2a$. Since the perimeter is $O=150 \text{~mm}$, by substituting into the formula $O=a+b+b$ we get $150=a+2a+2a$, so $5a=150$, from which we find $a=150:5=30 \text{~mm}$. Therefore, $b=2a=2 \cdot 30=60 \text{~mm}$.

b) The sides of the equilateral triangle are of equal length, i.e., $a=b=c$ and since its perimeter is $O=150 \text{~mm}$, by substituting into the formula $O=3a$, we get $3a=150$, i.e., $a=150:3=50 \text{~mm}$.