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Bisectors of a Triangle

A bisector is nothing but a ray or line that is used to partition an object into two equal parts. A triangle has a total of three bisectors. The bisectors of the triangle bisects the interior angles of the triangle. Since there are three angles in a triangle, the number of bisectors in a triangle is also three. In the above figure, the bisector of the triangle ABC is AD. This is because, the ray AD bisects the angle L A in to two equal parts.

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Properties of Bisectors of a Triangle Given below are some of the properties of bisectors of a triangle:

• Angle bisector of a triangle should be the line that bisects the inner angles.
• These bisectors should be intersect at center of the circle formed inside the triangle.
• The centre of the circle is called as incircle and is denoted by S in the figure.
• Every side of triangle should be cut by the corresponding angle bisector into two segments.
• Lengths of the angle bisectors is given by S$\alpha$, S$\beta$ and S$\gamma$.   • Radius of the incircle is calculated from the area of a triangle and is represented by r.

r = $\frac{2A}{(a + b + c)}$

Bisectors of a Triangle Solved Examples

Given below are some solved examples on bisectors of a triangle

Example 1: How to find the missing side length

bar(CA) = 18 bar(CD) = 24 bar(BA) =11 bar(DB) =? Solution: The given sides are bar(CA) = 18 bar(CD) = 24 bar(BA) = 11

So, we find out bar(DB)

Using theangle bisector theorem, bar(CA)/bar(CD) = bar(BA)/bar(DB)

=> 18/24 = 11/bar(DB)

18bar(DB) =264

bar(DB) = 14.66

Example 2: How to find the missing side length

bar(CA) =23bar(CD)=27bar(BA)=32bar(DB)=? Solution: The given sides are bar(CA) = 23 bar(CD) = 27 bar(BA) = 32

So, we find out bar(DB)

Using the angle bisector theorem, bar(CA)/bar(CD) = bar(BA)/bar(DB)

=> 23/27 = 32/bar(DB)

23bar(DB) = 864

bar(DB) = 37.56

Example 3: In a triangle, the three angles are given by 50, 14 and 24, these are the angle bisectors.Find an angle due to angle bisectors.

Solution: Step 1: 50° = 50/2 = 25°.

Step 2: 14° = 14/2 = 7°.

Step 3: 24° = 24/2 = 12°.

So, the angles are 25° , 7°, 12°.

Example 4: In a triangle, the sides of the triangle are 7, 3, and 12. Find the ratio of the distance from vertices.

Solution: Step 1: the given values are 7,3,12

a=7, b=3, c=12.

Step 2: Ratio of distance

a (b2 + c2 - a2) : b(c2 + a2 - b2) : c(a2 + b2 - c2).

Step 3: = 7 (32 + 122 - 72) : 3(122 +72 - 32) : 12(72 + 32 - 122).

Step 4: = 7(104):3(184):12(-86)

Step 5: = 728:552:-1032

Dividing by 4, we get, 182:138:-258

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