The measure of the central angle is the same as the measure of the intercepted arc. By the Inscribed Angle Theorem the measure of the inscribed angle is half the measure of its intercepted arc. This gives us the following relationship between a and b.
a = 1/2 b
Both a and b are positive, so this relationship tells us that a is less than b.
b We want to find the relationship between a and b from the following diagram.
The diagram shows a triangle with a and b as two of its sides. The angle opposite a has a measure of 62^(∘), but we do not know the measure of the angle opposite b. Once we know it, we can use the Law of Sines to relate a to b. So, first, let's find the measure of the last angle in our triangle. Let's call the measure of this angle x.
In a triangle, the measures of a the internal angles sum to 180^(∘). We can write an equation for x using this fact. Let's do so and solve the resulting equation.
We have that a = 1.188123... b. Since a and b are side lengths of a triangle, they are positive. Therefore, from a = 1.188123... b we conclude that a is greater than b.
c We want to find the relationship between a and b from the following diagram.
The diagram shows an equilateral triangle with a side length of 6 units and a square with a side length of 4 units. The area of the triangle is a, and the area of the square is b. We will find these areas, staring with the triangle.
Area of the Triangle
Let's recall the formula for an area A of an equilateral triangle with a side length s.
A = sqrt(3)/4 s^2
Let's substitute 6 for s in that formula and simplify.
