Internal Angle Bisector Theorem

Q.

In fig, $AD$ is the angle bisector of $\angle A$. If $BD=4cm,DC=3cm$ and $AB=6cm$, determine $AC$.

Q.

In $△$ABC, AB = 3 cm, AC = 4 cm and AD is the bisector of $\angle$A. Then, BD : DC is —

1. 9:16

2. 16:9

3. 3:4

4. 4:3

Q. If 𝑪𝑫 = 𝟑cm, 𝑫𝑩 = 𝟒 cm, 𝑩𝑭 = 𝟒cm, 𝑭𝑬 = 𝟓 cm, 𝑨𝑩 = 𝟔 cm and ∠𝑪𝑨𝑫 ≅ ∠𝑫𝑨𝑩 ≅ ∠𝑩𝑨𝑭 ≅ ∠𝑭𝑨𝑬, then the perimeter of quadrilateral 𝑨𝑬𝑩𝑪 is ___.

1. 26 cm
2. 35 cm
3. 28 cm
4. 30 cm

Q. The perimeter of $\Delta$ 𝑳𝑴𝑵 is 𝟑𝟐 𝐜𝐦. NX is the angle bisector of angle 𝑵, 𝑳𝑿 = 𝟑 𝐜𝐦, and 𝑿𝑴 = 𝟓 𝐜𝐦. Find 𝑳𝑵 and 𝑴𝑵.

1. LN = 9 cm and MN = 15 cm
2. LN = 15 cm and MN = 9 cm
3. LN = 12 cm and MN = 15 cm
4. None of the above

Q. In a Δ ABC, AD is the bisector of ∠A, meeting side BC at D. If BD = 2.5 cm, AB = 5 cm, and AC = 4.2 cm, find DC.

1. 3.2 cm
2. 2.1 cm
3. 2.5 cm
4. 5 cm

Q.

In $△ABC,AB=3cm,AC=4cm$ and AD is the bisector of $\angle$A. Then, $BD=DC.$

1. True

2. False

Q. In a Δ ABC, AD is the bisector of ∠A, meeting side BC at D. If BD = 2.5 cm, AB = 5 cm, and AC = 4.2 cm, find DC.

Q.

D, E, F are the mid points of the sides BC, CA and AB respectively of $△$ ABC. Then $△$ DEF is congruent to triangle –

1. ABC

2. AEF

3. BFD, CDE

4. AFE, BFD, CDE

Q.

In $△ABC,AB=3cm,AC=4cm$ and AD is the bisector of $\angle$A. Then, $BD:DC$ is

1. 9:16

2. 4:3

3. 16:9

4. 3:4

Q. If 𝑪𝑫 = 𝟑cm, 𝑫𝑩 = 𝟒 cm, 𝑩𝑭 = 𝟒cm, 𝑭𝑬 = 𝟓 cm, 𝑨𝑩 = 𝟔 cm and ∠𝑪𝑨𝑫 ≅ ∠𝑫𝑨𝑩 ≅ ∠𝑩𝑨𝑭 ≅ ∠𝑭𝑨𝑬, find the perimeter of quadrilateral 𝑨𝑬𝑩𝑪 .

Q.

D, E, F are the midpoints of the sides BC, CA and AB respectively of an equilateral $△ABC.$ Then $△AEF$ is congruent to triangle –

1. ABC

2. AEF

3. BFD, CDE

4. AFE, BFD, CDE

Q.

The perimeter of $△ABC$ is 49 cm. The angle bisector of $\angle A$ meets BC at M. BM = 6 cm, MC = 8 cm. Find AC.

1. 20 cm
2. 15 cm
3. 12 cm
4. 10 cm

Q. In $△ABC,AB=3cm,AC=4cm$ and AD is the bisector of $\angle$A. Then, $BD:DC$ is

1. 0.25

Q. In a Δ ABC, AD is the bisector of ∠A, meeting side BC at D. If BD = 2.5 cm, AB = 5 cm, and AC = 4.2 cm, find DC.

1. 2.5 cm
2. 3.2 cm
3. 5 cm
4. 2.1 cm

Q. In $△$ ABC, points P and Q are on CA and CB, respectively such that CA = 16 cm, AC = 10 cm, CB = 30 cm and CQ = 25 cm. PQ is parallel to AB or not?

Q. Show that any point on the bisector of an angle is equidistant from the arms of the angle.

Q. In triangle $ABC$, angle $B={90}^o$, $AB=y$ units, $BC=\sqrt{3}$ units, $AC=2$ units and angle $A=x^o$, find:
$\sin x^o$

Q. Construct a $△ABC$ in which $AB=6cm$, $\angle A={30}^{\circ }$ and $\angle B={60}^{\circ }$. Construct another $△A{B}^{\prime }{C}^{\prime }$ similar to $△ABC$ with base $A{B}^{\prime }=8cm$.

Q. In $△ABC,AB=3cm,AC=4cm$ and AD is the bisector of $\angle$A. Then, $BD:DC$ is

1. 0.25

Q. In a Δ ABC, AD is the bisector of ∠A, meeting side BC at D. If BD = 2.5 cm, AB = 5 cm, and AC = 4.2 cm, find DC.

Q. In $△ABC,AB=3cm,AC=4cm$ and AD is the bisector of $\angle$A. Is $BD=DC?$

Q. In the adjoining figure a $\Delta$ $ABC$ has been given in which $AD$ is its median $E$ is the midpoint of $AD,BE$ produced meets $AC$ in $F$ and $DGIEF$ meets $AC$ at $G$. If $AC=5.4$ cm then the length of $AF$ is:

1. $2.7$ cm
2. $3.6$ cm
3. $10.8$ cm
4. $1.8$ cm

Q. Fill in the blanks.
(i) Perpendicular bisector of the diameter of a circle passes through the P of the circle.
(ii) If B is image of A in line l and D is image of C in line l, then AC = Q .
(iii) Angle bisector is a ray which divides the angle in R equal parts.

1. P-Centre, Q-BD, R-2
2. P-Centre, Q-AD, R-1
3. P-Centre, Q-AB, R-1
4. P-Centre, Q-BC, R-2

Q.

If the areas of two similar triangles are equal, prove that they are congruent

Q. In a Δ ABC, AD is the bisector of ∠A, meeting side BC at D. If BD = 2.5 cm, AB = 5 cm, and AC = 4.2 cm, find DC.

1. 2.5 cm
2. 5 cm
3. 2.1 cm
4. 3.2 cm

Q. Match the following with their values if,
$△DGHand△DEF\text{are as shown below.}$

1. 7 units
2. 20 units
3. 30 units

Q. Match the following with their values if,
$△DGHand△DEF\text{are as shown below.}$

1. 7 units
2. 20 units
3. 30 units

Q. $△ABC$ and $△DBC$ are two isosceles triangles on the same base $BC$ and vertices $A$ and $D$ are on the same side of $BC$ as in figure.If $AD$ is extended to intersect $BC$ at $P,$Show that $AP$ bisects $\angle A$ as well as $\angle D$