If the sides of a triangle ABC are in A.P., then show that cot A 2 , cot B 2 , cot C 2 are also in A.P.

a , b , c a r e i n A . P d = common difreance a = b - d , c = b + d s = a + b + c 2 = b - d + b + b + d 2 = 3 b 2 c o t A 2 = s s - a s - b s - c = 3 b 2 3 b 2 - b + d 3 b 2 - b 3 b 2 - b - d = 3 b 2 b + 2 d 2 b 2 b - 2 d 2 c o t A 2 = 3 b + 2 d b - 2 d c o t B 2 = s s - b s - a s - c = 3 b 2 3 b - b 2 3 b 2 - b + d 3 b 2 = 3 b 2 b 2 b + 2 d 2 b - 2 d 2 c o t B 2 = 3 b 2 b 2 - 4 d 2 c o t c 2 = s s - c s - a s - b = 3 b 2 3 b 2 - b - d 3 b 2 - b + d 3 b 2 - b 3 b 2 b - 2 d 2 b + 2 d 2 b 2 c o t c 2 = 3 b - 2 d b + 2 d c o t A 2 + c o t c 2 = 3 b + 2 d b - 2 d + 3 b - 2 d b + 2 d = 3 b + 2 d + b - 2 d b - 2 d b + 2 d = 3 2 d b 2 - 4 d 2 = 2 3 . b 2 b 2 - 4 d 2 = 2 3 b 2 b 2 - 4 d 2 c o t A 2 + c o t c 2 = 2 c o t B 2 Hence, c o t A 2 , c o t B 2 a n d c o t c 2 a r e M . A . P

If the sides of a triangle ABC are in A.P., then show that cot A 2 , cot B 2 , cot C 2 are also in A.P.

a , b , c a r e i n A . P d = common difreance a = b - d , c = b + d s = a + b + c 2 = b - d + b + b + d 2 = 3 b 2 c o t A 2 = s s - a s - b s - c = 3 b 2 3 b 2 - b + d 3 b 2 - b 3 b 2 - b - d = 3 b 2 b + 2 d 2 b 2 b - 2 d 2 c o t A 2 = 3 b + 2 d b - 2 d c o t B 2 = s s - b s - a s - c = 3 b 2 3 b - b 2 3 b 2 - b + d 3 b 2 = 3 b 2 b 2 b + 2 d 2 b - 2 d 2 c o t B 2 = 3 b 2 b 2 - 4 d 2 c o t c 2 = s s - c s - a s - b = 3 b 2 3 b 2 - b - d 3 b 2 - b + d 3 b 2 - b 3 b 2 b - 2 d 2 b + 2 d 2 b 2 c o t c 2 = 3 b - 2 d b + 2 d c o t A 2 + c o t c 2 = 3 b + 2 d b - 2 d + 3 b - 2 d b + 2 d = 3 b + 2 d + b - 2 d b - 2 d b + 2 d = 3 2 d b 2 - 4 d 2 = 2 3 . b 2 b 2 - 4 d 2 = 2 3 b 2 b 2 - 4 d 2 c o t A 2 + c o t c 2 = 2 c o t B 2 Hence, c o t A 2 , c o t B 2 a n d c o t c 2 a r e M . A . P

In a triangle P Q R , let ∠ P Q R = 30 ° and the sides P Q and Q R have lengths 10 3 a n d 10 units, respectively. Then, which of the following statement(s) is (are) TRUE?

The area of the circumcircle of the triangle P Q R is 100 π sq. units

HARD

Earn 100

If the sides of a triangle $ABC$ are in A.P., then show that cot $\frac{A}{2}, \cot \frac{B}{2}, \cot \frac{C}{2}$ are also in A.P.

Important Questions on Properties of Triangles

HARD

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

In a non-right-angled triangle

Δ P Q R,

let

p, q, r

denote the lengths of the sides opposite to the angles at

P, Q, R

respectively. The median from

R

meets the side

P Q

S,

the perpendicular from

P

meets the side

Q R

E

, and

R S

and

P E

intersect at

O .

p = \sqrt{3}, q = 1,

and the radius of the circumcircle of the

Δ P Q R

equals

1,

then which of the following options is/are correct?

HARD

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

Let

A B C

be an acute angled triangle and let D be the midpoint of BC. If

A B = A D,

then

\frac{\tan (B)}{\tan (C)}

equals

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

If the lengths of the sides of a triangle are in A.P and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is:

HARD

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

Let

A B C

be an acute-angled triangle and let

D

be the mid-point of

B C

. If

A B = A D

, then

\frac{\tan B}{\tan C}

equal

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

In a triangle

A B C

with

∠ A = 90 °

P

is a point on

B C

such that

P A : P B = 3 : 4 .

A B = \sqrt{7} and A C = \sqrt{5}

then

B P : P C

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

a, b, c

are in

GP

and

\log a - \log 2 b, \log 2 b - \log 3 c

and

\log 3 c - \log a

are in

AP

, then

a, b

and

c

are the lengths of the sides of a triangle, which is

HARD

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

In a triangle the sum of two sides is

x

and the product of the same two sides is

y

. If

x^{2} - c^{2} = y,

(where

c

is the third side of the triangle) then the ratio of the inradius to the circumradius of the triangle is

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

In a triangle

A B C

, if

A = 2 B

and the sides opposite to the angles

A, B, C

are

α + 1, α - 1

and

α

respectively, then

α =

HARD

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

In a

∆ P Q R

P

is the largest angle and

\cos P = \frac{1}{3}

. Further the incircle of the triangle touches the sides

P Q, Q R

and

R P

N, L

and

M

respectively, such that the lengths of

P N, Q L

and

R M

are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

HARD

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

The lengths of two adjacent sides of a cyclic quadrilateral are

2

units and

5

units and the angle between them is

60^{o}

. If the area of the quadrilateral is

4 \sqrt{3}

sq. units, then the perimeter of the quadrilateral is

EASY

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

In any triangle

A B C

(b + c) \cos A + (c + a) \cos B + (a + b) \cos C

equals

HARD

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

In a triangle

P Q R,

let

∠ P Q R = 30 °

and the sides

P Q

and

Q R

have lengths

10 \sqrt{3} a n d 10

units, respectively. Then, which of the following statement(s) is (are) TRUE?

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

Let

a, b

and

c

be the lengths of the sides of a triangle with its opposite angles

A, B

and

C

respectively. If

a = 3, b = 4

and

A = \sin^{- 1} (\frac{3}{4}),

then the angle

B

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

The angles

A, B & C

of a

∆ A B C

are in

A . P .

and

a : b = 1 : \sqrt{3} .

c = 4 c m,

then the area (in

s q . c m

) of this triangle is:

HARD

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

In a

∆ A B C

, if

b = 10

a \cos^{2} \frac{C}{2} + c \cos^{2} \frac{A}{2} = 15

and the area of the triangle is

15 \sqrt{3}

sq. units, then

\cot \frac{B}{2} =

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

△ A B C

, if

a = 5

and

\tan \frac{A - B}{2} = \frac{1}{4} \tan \frac{A + B}{2}

, then

\sqrt{a^{2} - b^{2}} =

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

Δ A B C

, right-angled at

A

, the circumradius, inradius and radius of the excircle opposite to

A

are respectively in the ratio

2 : 5 : λ

, then the roots of the equation

x^{2} - (λ - 5) x + (λ - 6) = 0

are

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

Let

A B C

be a triangle such that

A B = B C .

Let

F

be the midpoint of

A B

and

X

be a point on

B C

such that

F X

is perpendicular to

A B .

B X = 3 X C

then the ratio

\frac{B C}{A C}

equals,

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

Δ A B C

, if

\sin^{2} A + \sin^{2} B = \sin^{2} C

and

l (A B) = 10

, then the maximum value of the area of

Δ A B C

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Half-Angle Formula and the Area of a Triangle

With the usual notation in

∆ A B C

, if

∠ A + ∠ B = 120 °, a = \sqrt{3} + 1

units and

b = \sqrt{3} - 1

units, then the ratio

∠ A : ∠ B

If the sides of a triangle ABC are in A.P., then show that cot A2,cotB2,cotC2 are also in A.P.

Important Questions on Properties of Triangles

Learn and Prepare for any exam you want !

If the sides of a triangle $ABC$ are in A.P., then show that cot $\frac{A}{2}, \cot \frac{B}{2}, \cot \frac{C}{2}$ are also in A.P.