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

In $∆ A B C$ , the tangent of half the difference of two interior angles is one-third the tangent of half the sum of these angles. Then, the ratio of the side opposite to smaller angle between these to the side opposite to large angle between these is equal to

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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

In ∆ABC, the tangent of half the difference of two interior angles is one-third the tangent of half the sum of these angles. Then, the ratio of the side opposite to smaller angle between these to the side opposite to large angle between these is equal to

Important Questions on Properties of Triangles

Learn and Prepare for any exam you want !

In $∆ A B C$ , the tangent of half the difference of two interior angles is one-third the tangent of half the sum of these angles. Then, the ratio of the side opposite to smaller angle between these to the side opposite to large angle between these is equal to