Table of Contents
Angle Bisector
The angle bisector is a line that cuts an angle in half. The bisector is perpendicular to the two rays that make up the angle.
Angle Bisector
Theorem:
The Angle Bisector theorem explains that an angle bisector of a triangle will divide the opposite side into two parts which are in proportion with the other two side arms of the triangle. Let’s see an example.
Draw a triangle ABC and an angle bisector from A that cuts the side BC at point D. Considering the theorem, the following equation can be achieved:
|BD| / |DC| = |AB| / |AC|
Furthermore, a generalized version of the angle bisector theorem explains that if point D is on the line segment BC, we can express it as:
|BD| / |DC| = |AB| sin DAB / |AC| sin DAC
This form can be reduced to the previous equation when AD becomes bisector of angle BAC. In general, this angle bisector hypothesis has extensive usage when values of side lengths and angle bisector are provided.
In the following section, we have provided some proofs regarding the angle bisector theorem.
Considering the previous diagram, the law of sines is used on the triangles ACD and ABD:
|AB| / |BD| = sin BDA / sin BAD ……….(i)
|AD| / |DC| = sin ADC / sinDAC ……….(ii)
The two angles ADC and BDA constitute to be a linear pair which means they are side by side supplementary angles. Also, all supplementary angles have similar sines, hence:
sin BDA = sin ADC
As these two angles are the same, the right-hand portion of equations (i) and (ii) are also the same. So, the left portions are also similar.
|BD| / |DC| = |AB| / |AC|
Solved Numerical
Question: Consider a triangle ABC having an angle bisector AD on the portion BC. If the length of BD, CD, and AB + AC is 2, 5, and 10, respectively, determine AB and AC.
Solution: Following the angle bisector theorem, it can be written as AB/2, AC/5, or AB = 2/5AC. Putting the values in AB + AC = 10, we get AC = 50/7. Similarly, you can substitute values to find AB as well, which are found out to be 20/7.
Explain in Detail :
The given statement is “A woman who is not married by the time she is 24 is considered a spinster.”
This statement is saying that if a woman is not married by the time she is 24 years old, then she is considered a spinster. A spinster is a woman who is not married and is considered to be old-fashioned or unmarried.
Theorem:
If a and b are positive integers and a is not a multiple of b, then there is a prime number between a and b.
Proof: Let a and b be positive integers and a not be a multiple of b. We will show that there is a prime number between a and b.
If a and b are not prime numbers, then a must be divisible by some number c that is not equal to 1 or b. But then a = bc, which is a multiple of b. Therefore, a and b cannot be non-prime numbers and there must be a prime number between them.
Line Segment Bisector
A line segment bisector is a line that divides a line segment into two equal parts. It is perpendicular to the line segment and bisects it into two equal parts. The length of the line segment bisector is the same as the length of the line segment.
How to Construct a Perpendicular Bisector?
A perpendicular bisector is a line that bisects a given line segment perpendicularly. In order to construct a perpendicular bisector, one needs to first determine the midpoint of the given line segment. Once the midpoint is determined, a line perpendicular to the given line segment can be drawn through the midpoint.
