As every the Angle Bisector theorem, the angle bisector the a triangle bisects the opposite next in such a means that the ratio of the 2 line-segments is proportional come the ratio of the other two sides. For this reason the relative lengths of the opposite side (divided by edge bisector) are equated to the lengths of the various other two political parties of the triangle. Edge bisector to organize is applicable come all types of triangles. 

Class 10 students deserve to read the concept of angle bisector theorem here together with the proof. Personally from angle bisector theore, we will also discuss here the external angle theorem, perpendicular bisector theorem, converse of edge bisector theorem.

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Table the Contents:

What is angle Bisector Theorem?

An edge bisector is a straight line drawn from the vertex of a triangle come its opposite next in together a way, the it divides the angle right into two same or congruent angles. Now let united state see, what is the edge bisector theorem.

According to the edge bisector theorem, the edge bisector the a triangle divides the opposite side into two parts that are proportional to the other two sides of the triangle.

Interior angle Bisector Theorem

In the triangle ABC, the angle bisector intersects next BC at allude D. View the figure below.

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As per the edge bisector theorem, the ratio of the heat segment BD come DC equals the ratio of the length of the side abdominal muscle to AC.

(frac BD ight =frac AC ight )

Conversely, once a suggest D top top the next BC divides BC in a ratio similar to the political parties AC and AB, climate the angle bisector the ∠ A is AD. Hence, according to the theorem, if D lies top top the next BC, then,

(fracleft left =fracleft left )

If D is external to the next BC, directed angles and also directed line segments are forced to be applied in the calculation.

Angle bisector organize is used when next lengths and also angle bisectors room known.

Proof of edge bisector theorem

We can conveniently prove the angle bisector theorem, by utilizing trigonometry here. In triangle ABD and ACD (in the above figure) using regulation of sines, we deserve to write;

(fracABBD=fracsinangle BDAsinangle BAD) ….(1)

(fracACDC=fracsinangle ADCsinangle DAC) ….(2)

The angles ∠ ADC and ∠ BDA do a straight pair and hence called surrounding supplementary angles. 

Since the sine that supplementary angles are equal, therefore,

Sin ∠ BDA = Sin ∠ ADC …..(3)

Also, 

∠ DAC = ∠ poor (AD is the angle bisector)

Thus, 

Sin ∠ BDA = Sin ∠ ADC …(4)

Hence, native equation 3 and 4, we have the right to say, the RHS that equation 1 and also 2 space equal, therefore, LHS will also be equal.

(frac=fracleft )

Hence, angle bisector to organize is proved. 

Condition:

If the angle ∠ DAC and also ∠ BAD are not equal, the equation 1 and also equation 2 deserve to be written as:

(fracleft ) sin ∠ negative = sin∠ BDA

(frac DC ight ) sin ∠ DAC = sin∠ ADC

Angles ∠ ADC and also ∠ BDA room supplementary, therefore the RHS of the equations are still equal. Hence, we get

(fracleft ) sin ∠BAD = (fracleft left ) sin ∠DAC

This rearranges to generalized view the the theorem.

Converse of edge Bisector Theorem

In a triangle, if the interior suggest is equidistant indigenous the 2 sides that a triangle then that allude lies top top the angle bisector the the angle developed by the two line segments.

Triangle edge Bisector Theorem

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Extend the next CA to accomplish BE to meet at suggest E, such the BE//AD.

Now we can write,

CD/DB = CA/AE (since AD//BE) —-(1)

∠4 = ∠1

∠1 = ∠2

∠2 = ∠3

∠3 = ∠4

ΔABE is one isosceles triangle with AE=AB 

Now if we change AE by abdominal muscle in equation 1, we get;

CD/DB = CA/AB

Hence proved.

Perpendicular Bisector Theorem

According to this theorem, if a point is equidistant indigenous the endpoints the a heat segment in a triangle, climate it is on the perpendicular bisector of the line segment. 

Alternatively, we have the right to say, the perpendicular bisector bisects the offered line segment into two equal parts, come which that is perpendicular. In instance of triangle, if a perpendicular bisector is drawn from the vertex come the the opposite side, climate it divides the segment into two congruent segments.

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In the above figure, the line segment SI is the perpendicular bisector of WM.

External edge Bisector Theorem

The external angle bisector the a triangle divides the opposite next externally in the ratio of the political parties containing the angle. This problem occurs commonly in non-equilateral triangles.

Proof

Given : In ΔABC, ad is the exterior bisector of ∠BAC and also intersects BC created at D. 

To prove : BD/DC = AB/AC

Constt: draw CE ∥ DA meeting ab at E

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Since, CE ∥ DA and also AC is a transversal, therefore, 

∠ECA = ∠CAD (alternate angles) ……(1)

Again, CE ∥ DA and BP is a transversal, therefore,

∠CEA = ∠DAP (corresponding angles) —–(2)

But advertisement is the bisector that ∠CAP, 

∠CAD = ∠DAP —–(3)

As us know, political parties opposite to equal angles space equal, therefore,

∠CEA = ∠ECA

In ΔBDA, EC ∥ AD. 

BD/DC = BA/AE

AE = AC, 

BD/DC = BA/AC

Hence, proved.

Solved instances on angle Bisector Theorem

Go through the following examples to understand the concept of the edge bisector theorem.

Example 1:

Find the value of x because that the offered triangle utilizing the edge bisector theorem.

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Solution:

Given that,

AD = 12, AC = 18, BC=24, DB = x

According to edge bisector theorem, 

AD/AC = DB/BC

Now substitute the values, us get

12/18 = x/24

X = (⅔)24

x = 2(8)

x= 16

Hence, the value of x is 16.

Example 2:

ABCD is a square in i beg your pardon the bisectors of edge B and also angle D intersects top top AC at allude E. Display that AB/BC = AD/DC

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Solution:

From the offered figure, the segment DE is the edge bisector of angle D and BE is the internal angle bisector of edge B.

Hence, the using interior angle bisector theorem, us get

AE/EC = AD/DC ….(1)

Similarly, 

AE/EC = AB/BC ….(2)

From equations (1) and (2), we get

 AB/BC = AD/DC

Hence, AB/BC = AD/DC is proved.

Example 3.

In a triangle, AE is the bisector of the exterior ∠CAD that meets BC at E. If the value of ab = 10 cm, AC = 6 cm and BC = 12 cm, discover the value of CE.

Solution:

Given : abdominal = 10 cm, AC = 6 cm and also BC = 12 cm

Let CE is equal to x.

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By exterior angle bisector theorem, we know that,

BE / CE = ab / AC

(12 + x) / x = 10 / 6

6( 12 + x ) = 10 x < by overcome multiplication>

72 + 6x = 10x

72 = 10x – 6x

72 = 4x

x = 72/4

x = 18

CE = 18 cm

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