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 angleRelated Calculators | |

Area of Triangle | Triangle Calculator |

Area of a Equilateral Triangle Calculator | Area of a Right Triangle Calculator |

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

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 (b^{2} + c^{2} - a^{2}) : b(c^{2} + a^{2} - b^{2}) : c(a^{2} + b^{2} - c^{2}).

Step 3: = 7 (3^{2} + 12^{2} - 7^{2}) : 3(12^{2} +7^{2} - 3^{2}) : 12(7^{2} + 3^{2} - 12^{2}).

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

Step 5: = 728:552:-1032

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

Related Topics | |

Math Help Online | Online Math Tutor |