In a △ A B C express ∑ r 1 cot A 2 in terms of s .

Given, ∑ r 1 cot A 2 = r 1 cot A 2 + r 2 cot B 2 + r 3 cot C 2 We know that, r 1 = s tan A 2 , r 2 = s tan B 2 , r 3 = s tan C 2 = s tan A 2 cot A 2 + s tan B 2 cot B 2 + s tan C 2 cot C 2 = s + s + s ∴ ∑ r 1 cot A 2 = 3 s

In a △ A B C show that ∑ a cot A = 2 R + r .

To show, ∑ a cot A = 2 ( R + r ) LHS = ∑ a cot A = a cot A + b cos B + c cot C = 2 R sin A cos A sin A + 2 R sin B cos B sin B + 2 R sin C cos C sin C [From sine rule] = 2 R { cos A + cos B + cos C } = 2 R 2 cos A + B 2 cos A - B 2 + 1 - 2 sin 2 C 2 = 2 R 1 + 2 cos π - C 2 cos A - B 2 - 2 sin 2 C 2 [since A + B + C = π ] = 2 R 1 + 2 sin C 2 cos A - B 2 - 2 sin 2 C 2 [since cos π 2 - x = sin x ] = 2 R 1 + 2 sin C 2 cos A - B 2 - sin C 2 = 2 R 1 + 2 sin C 2 cos A - B 2 - cos A + B 2 [since A + B + C 2 = 90 ° ] = 2 R 1 + 2 sin C 2 × 2 sin A + B 2 + A - B 2 2 sin A + B 2 - A - B 2 2 = 2 R 1 + 4 sin A 2 sin B 2 sin C 2 = 2 R 1 + 4 R sin A 2 sin B 2 sin C 2 R = 2 R 1 + r R = 2 { R + r } = RHS Hence, proved.

In △ A B C , prove that r 1 + r 2 + r 3 - r = 4 R .

To prove, r 1 + r 2 + r 3 - r = 4 R We know that, r 1 = 4 R sin A 2 cos B 2 cos C 2 , r 2 = 4 R cos A 2 sin B 2 cos C 2 , r 3 = 4 R cos A 2 cos B 2 sin C 2 and r = 4 R sin A 2 sin B 2 sin C 2 LHS = r 1 + r 2 + r 3 - r = 4 R sin A 2 cos B 2 cos C 2 + 4 R cos A 2 sin B 2 cos C 2 + 4 R cos A 2 cos B 2 sin C 2 - 4 R sin A 2 sin B 2 sin C 2 = 4 R cos C 2 sin A 2 cos B 2 + cos A 2 sin B 2 + 4 R sin C 2 cos A 2 cos B 2 - sin A 2 sin B 2 = 4 R cos C 2 sin A 2 + B 2 + 4 R sin C 2 cos A 2 + B 2 = 4 R cos C 2 sin 90 ° - C 2 + 4 R sin C 2 cos 90 ° - C 2 [since A + B + C = 180 ° ] = 4 R cos C 2 cos C 2 + 4 R sin C 2 sin C 2 = 4 R cos 2 C 2 + 4 R sin 2 C 2 = 4 R cos 2 C 2 + sin 2 C 2 = 4 R = RHS Hence, proved.

HARD

11th Telangana Board

IMPORTANT

Earn 100

A lamp post is situated at the middle point $M$ of the side $A C$ of a triangular plot $A B C$ with $B C = 7 m$ , $C A = 8 m$ and $A B = 9 m$ . Lamp post subtends an angle $15 °$ at the point $B$ . Find the height of the lamp post.

Important Questions on Properties of Triangles

HARD

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 10 - Properties of Triangles>Exercise 10 (a)>Q III

Two ships leave a port at the same time. One goes

24 km

per hour in the direction

N 45 ° E

and other travels

32 km

per hour in the direction

S 75 ° E

. Find the distance between the ships at the end of

3

hours.

HARD

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 10 - Properties of Triangles>Exercise 10 (a)>Q III

A tree stands vertically on the slant of the hill. From a point

A

on the ground

35

meters down the hill from the base of the tree, the angle of elevation of the top of the tree is

60 °

. If the angle of elevation of the foot of the tree from

A

15 °

, then find the height of the tree.

HARD

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 10 - Properties of Triangles>Exercise 10 (a)>Q III

The upper

{\frac{3}{4}}^{th}

portion of a vertical pole subtends an angle

\tan^{- 1} \frac{3}{5}

at a point in the horizontal plane through its foot and at a distance

40 m

from the foot. Given that the vertical pole is at a height less than

100 m

from the ground, find its height.

MEDIUM

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 10 - Properties of Triangles>Exercise 10 (a)>Q III

A B

is a vertical pole with

B

at the ground level and

A

at the top. A man finds that the angle of elevation of the point

A

from a certain point

C

on the ground is

60 °

. He moves away from the pole along the line

B C

to a point

D

such that

C D = 7 m

. From

D

, the angle of elevation of the point

A

45 °

. Find the height of the pole.

HARD

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 10 - Properties of Triangles>Exercise 10 (a)>Q III

Let an object be placed at some height

h cm

and let

P

and

Q

be two points of observation which are at a distance

10 cm

apart on a line inclined at angle

15 °

to the horizontal. If the angles of elevation of the object from

P

and

Q

are

30 °

and

60 °

respectively, then find

h

EASY

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 10 - Properties of Triangles>Exercise 10 (b)>Q I

In a

△ A B C

express

\sum_{}^{} r_{1} \cot \frac{A}{2}

in terms of

s

HARD

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 10 - Properties of Triangles>Exercise 10 (b)>Q I

In a

△ A B C

show that

\sum_{}^{} a \cot A = 2 (R + r)

MEDIUM

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 10 - Properties of Triangles>Exercise 10 (b)>Q I

△ A B C

, prove that

r_{1} + r_{2} + r_{3} - r = 4 R

A lamp post is situated at the middle point M of the side AC of a triangular plot ABC with BC=7m, CA=8m and AB=9m. Lamp post subtends an angle 15° at the point B. Find the height of the lamp post.

Important Questions on Properties of Triangles

Learn and Prepare for any exam you want !

A lamp post is situated at the middle point $M$ of the side $A C$ of a triangular plot $A B C$ with $B C = 7 m$ , $C A = 8 m$ and $A B = 9 m$ . Lamp post subtends an angle $15 °$ at the point $B$ . Find the height of the lamp post.