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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.

Bisectors of a Triangle

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

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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
Radius
Radius

  • 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

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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) =?`

Finding Missing Side of Triangle

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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