Law of Cosines
Solved Problems
Example 1.
Find the least diagonal of a regular hexagon with side length
Solution.
The internal angle in a regular hexagon is
Then the least diagonal is equal to
Example 2.
Two sides of a triangle are equal to
Solution.
By the Cosine Rule, we have
Calculate
Hence,
Example 3.
The middle side of a triangle is
Solution.
Let
In any triangle, the mid-sized side and mid-sized angle are opposite to each other. Hence, using the Law fo Cosines, we can write:
Solve this equation for
The negative root
Note that this triangle satisfies the Pythagorean theorem:
In such a case, the area of the triangle is
Example 4.
In a triangle
Solution.
The midline
Let
Recall that
Find the discriminant of the quadratic equation:
The roots of the equation are
We choose the positive root, so
Example 5.
Given a triangle whose sides are
Solution.
Use the Law of Cosines for the triangle
Solve this equation for
Apply now the Law of Cosines to the triangle
Subctitute
Example 6.
Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.
Solution.
Consider a parallelogram
Denote the angle
Applying the Law of Cosines to the triangle
By the Reduction Identity,
This yields:
Similarly we find the square of the diagonal
or
Calculate the sum of the squares of the diagonals:
The right-hand side represents the sum of the squares of four sides of the parallelogram.