MEDIUM

Earn 100

State and prove the converse of angle bisector theorem

Important Questions on Triangles

HARD

Mathematics>Geometry>Triangles>Angle Bisector Theorem

The angle bisectors

B D

and

C E

of a

Δ A B C

are divided by the incentre

I

in the ratios

3 : 2

and

2 : 1

respectively. Then, the ratio in which

I

divides the angle bisector through

A

MEDIUM

Mathematics>Geometry>Triangles>Angle Bisector Theorem

Let

P Q R

be an acute-angled triangle in which

P Q < Q R

. From the vertex

Q

draw the altitude

Q Q_{1}

the angle bisector

Q Q_{2}

and the median

Q Q_{3}

, with

Q_{1,} Q_{2,} Q_{3}

lying on

P R

. Then,

MEDIUM

Mathematics>Geometry>Triangles>Angle Bisector Theorem

In a triangle

A B C, ∠ B A C = 90 °; A D

is the altitude from

A

on to

B C

. Draw

D E

perpendicular to

A C

and

D F

perpendicular to

A B

. Suppose

A B = 15

and

B C = 25 .

Then the length of

E F

MEDIUM

Mathematics>Geometry>Triangles>Angle Bisector Theorem

∆ A B C

\vec{A D}

is the bisector of

∠ A

intersects

\bar{B C}

D

∴ B D =

_____

MEDIUM

Mathematics>Geometry>Triangles>Angle Bisector Theorem

a, b, c

are lengths of the sides

B C, C A

and

A B

respectively of

Δ A B C

and

H

is any point in the plane of

∆ A B C

such that

a \vec{A H} + b \vec{B H} + c \vec{C H} = \vec{0}

, then

H

is the

HARD

Mathematics>Geometry>Triangles>Angle Bisector Theorem

Question Image

In $△ A B C$ , it is given that $\frac{B D}{D C} = \frac{A B}{A C}, ∠ B = 70 °, ∠ C = 50 ° .$ Find $∠ B A D$ in degrees.

EASY

Mathematics>Geometry>Triangles>Angle Bisector Theorem

Consider a

Δ P Q R

in which the relation

Q R^{2} + P R^{2} = 5 P Q^{2}

holds. Let

G

be the point of intersection of medians

P M

and

Q N

. Then,

∠ Q G M

is always

MEDIUM

Mathematics>Geometry>Triangles>Angle Bisector Theorem

In the given fig. $A M ⊥ B C$ and $A N$ is the bisector of $∠ A$ . If $∠ B = 65 °$ and $∠ C = 33 °$ , then the value of $∠ M A N$ will be

Question Image

EASY

Mathematics>Geometry>Triangles>Angle Bisector Theorem

P (0, 0), Q (1, 0)

and

R (\frac{1}{2}, \frac{\sqrt{3}}{2})

are three given points, then the centre of the circle for which the lines

P Q, Q R

and

R P

are the tangents is

EASY

Mathematics>Geometry>Triangles>Angle Bisector Theorem

If the incentre and the circumcentre of the triangle formed by the lines

x = 2

4 x + 3 y + 7 = 0

and

y = 3

are

I

and

S

respectively, then

I S =

EASY

Mathematics>Geometry>Triangles>Angle Bisector Theorem

In $△ P Q R$ , the bisector of $∠ P$ intersects $Q R$ in $M$ . If $P Q = 10, P R = 12, Q M = 8$ then $Q R =$ _____.

MEDIUM

Mathematics>Geometry>Triangles>Angle Bisector Theorem

The incentre of the triangle with vertices

(1, \sqrt{3}), (0, 0)

and

(2, 0)

is:

MEDIUM

Mathematics>Geometry>Triangles>Angle Bisector Theorem

D

and

E

are the points on sides

B C

and

A C

, respectively of

∆ A B C . A D

and

B E

intersect each other at T. If

\frac{A T}{T D} = 5

and

\frac{B T}{E T} = 7

, then

C D : B D =

HARD

Mathematics>Geometry>Triangles>Angle Bisector Theorem

If the line

3 x + 4 y - 24 = 0

intersects the

x

-axis at the point

A

and the

y

-axis at the point

B,

then the incentre of the triangle

O A B,

where

O

is the origin, is:

EASY

Mathematics>Geometry>Triangles>Angle Bisector Theorem

∆ ABC

∠ ABC = 90 °

and

∠ BAC = 60 °

. If bisector of

∠ BAC

meets

BC

D

. Then

BD : DC

MEDIUM

Mathematics>Geometry>Triangles>Angle Bisector Theorem

A B C D

is a rectangle and

P

is a point inside it such that

A P = 33, B P = 16, D P = 63 .

Find

C P .

EASY

Mathematics>Geometry>Triangles>Angle Bisector Theorem

In the given figure, $A D$ is the bisector of $∠ B A C$ . If $A B = 10 c m, A C = 6 c m$ and $B C = 12 c m$ . Find $B D$ .

Question Image

EASY

Mathematics>Geometry>Triangles>Angle Bisector Theorem

Let $a, b, c$ be the side-length of a triangle and $l, m, n$ be the lengths of its medians. Put $K = \frac{l + m + n}{a + b + c}$ Then, as $a, b, c$ vary, $K$ can assume every value in the interval

MEDIUM

Mathematics>Geometry>Triangles>Angle Bisector Theorem

The incentre of the triangle formed by the straight line having

3

X

-intercept and

4

Y

-intercept, together with the coordinate axes, is

MEDIUM

Mathematics>Geometry>Triangles>Angle Bisector Theorem

The

x -

coordinate of the incentre of the triangle that has the coordinates of midpoints of its sides as

(0, 1), (1, 1)

and

(1, 0)

State and prove the converse of angle bisector theorem

Important Questions on Triangles

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