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CAT Angle Bisector Theorem - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

Quick Facts

  • 10 Questions around this concept.

Solve by difficulty

In the given figure, if BD:CD = 4:5, ∠BAD = ∠CAD, and AB = 12 cm, then find the length of AC.

In ABC, it is given that \frac{A B}{A C}=\frac{B D}{D C}. If ∠ B = 70° and ∠ C = 50° then ∠ BAD is

In the figure, \overline{A B} \perp \overline{C D}, and AD is the bisector of ∠BAE. AB = 4 cm and AC = 5 cm. Find CD.

In the given figure, if BD:CD = 4:5, ∠BAD = ∠CAD, and AB = 12 cm, then find the length of AC.

Concepts Covered - 1

Angle Bisector Theorem

Theorem: The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

Let us prove this.

Given: In ΔABC, AD is the bisector of ∠A, and the bisector of ∠A meets BC in D.

We need to prove: \mathrm{\frac{B D}{D C}=\frac{A B}{A C}}.

Construction: Draw CP parallel to AD to meet BA produced at P.

 Proof:

∠DAC = ∠ACP                  (alternate angles and AD || CP)

∠BAD = ∠APC                  (corresponding angles)

But ∠BAD = ∠D                (given)

∴ ∠ACP = ∠APC

In triangle APC,

AC = AP                             (sides opposite to equal angles are equal) In triangle BCP,

\mathrm{\frac{B D}{D C}=\frac{B A}{A P}}                    (by basic proportionality theorem)

\Rightarrow \mathrm{\frac{B D}{D C}=\frac{B A}{A C} \quad(\because A P=A C)}

Hence Proved.

The converse of Angle Bisector Theorem

If a line that passes through a vertex of a triangle, divides the base in the ratio of the other two sides, then it bisects the angle.

In the figure, AD divides BC in the ratio \mathrm{\frac{B D}{D C}} and if \mathrm{\frac{B D}{D C}=\frac{A B}{A C}} then AD is the bisector of ∠A.

 

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