Question:
Find the circumradius if 2 is the side of the triangle with the opposite angle \(\frac{\pi}{3}\).
Find the circumradius if 2 is the side of the triangle with the opposite angle \(\frac{\pi}{3}\).
Updated On: Aug 18, 2023
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Solution and Explanation
To find the circumradius of a triangle, we can use the formula:
R = \(\frac{a}{2sin(A)}\)
Where R is the circumradius, a is the length of a side of the triangle, and A is the opposite angle.
In this case, we have the side length a = 2 and the opposite angle A = \(\frac{\pi}{3}\). Substituting these values into the formula, we get:
R = \(\frac{2}{2sin(\frac{\pi}{3})}\)
Now, we can simplify the expression:
R = \(\frac{2}{(2\times\frac{\sqrt{}3}{2})}\)
R =\(\frac{2}{(\sqrt3)}\)
R = \(\frac{2\sqrt3}{3}\)
Therefore, the circumradius of the triangle is \(\frac{2\sqrt3}{3}\).
R = \(\frac{a}{2sin(A)}\)
Where R is the circumradius, a is the length of a side of the triangle, and A is the opposite angle.
In this case, we have the side length a = 2 and the opposite angle A = \(\frac{\pi}{3}\). Substituting these values into the formula, we get:
R = \(\frac{2}{2sin(\frac{\pi}{3})}\)
Now, we can simplify the expression:
R = \(\frac{2}{(2\times\frac{\sqrt{}3}{2})}\)
R =\(\frac{2}{(\sqrt3)}\)
R = \(\frac{2\sqrt3}{3}\)
Therefore, the circumradius of the triangle is \(\frac{2\sqrt3}{3}\).
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Concepts Used:
Coordinate Geometry
Coordinate geometry, also known as analytical geometry or Cartesian geometry, is a branch of mathematics that combines algebraic techniques with the principles of geometry. It provides a way to represent geometric figures and solve problems using algebraic equations and coordinate systems.
The central idea in coordinate geometry is to assign numerical coordinates to points in a plane or space, which allows us to describe their positions and relationships using algebraic equations. The most common coordinate system is the Cartesian coordinate system, named after the French mathematician and philosopher René Descartes.
The central idea in coordinate geometry is to assign numerical coordinates to points in a plane or space, which allows us to describe their positions and relationships using algebraic equations. The most common coordinate system is the Cartesian coordinate system, named after the French mathematician and philosopher René Descartes.