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Suppose we have two circles of radius $2$ each in the plane such that the distance between their centres is $2 \sqrt{3}$ . The area of the region common to both circles lies between

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Important Questions on Properties of Triangle

MEDIUM

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

If in a triangle

A B C, a^{2} + 2 b c - (b^{2} + c^{2}) = a b \sin \frac{C}{2} \cos \frac{C}{2}

, then

\cot (B + C) =

HARD

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

In the figure given below, if the areas of the two regions are equal then which of the following is true?

Question Image

HARD

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

Let

x, y

and

z

be positive real numbers. Suppose

x, y

and

z

are the lengths of the sides of a triangle opposite to its angles

X, Y

and

Z

respectively. If

\tan \frac{X}{2} + \tan \frac{Z}{2} = \frac{2 y}{x + y + z}

, then which of the following statements is/are TRUE?

MEDIUM

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

ΔABC,

the lengths of sides

A C

and

A B

are

12 cm

and

5 cm,

respectively. If the area of

∆ ABC

30 {cm}^{2}

and

R

and

r

are respectively the radii of circumcircle and incircle of

ΔABC

, then the value of

2 R + r (in cm)

is equal to ______ .

HARD

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

Let

A B C D

be a square and let

P

be point on segment

C D

such that

D P : P C = 1 : 2

. Let

Q

be a point on segment

A P

such that

∠ B Q P = 90^{o}

. Then the ratio of the area of quadrilateral

P Q B C

to the area of the square

A B C D

HARD

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

Let

A B C

be a triangle such that

A B = 4, B C = 5

and

C A = 6

. Choose points

D, E, F

A B, B C, C A

respectively, such that

A D = 2, B E = 3, C F = 4

. Then

\frac{area Δ D E F}{area Δ A B C}

MEDIUM

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

In a $Δ A B C$ , points $X$ and $Y$ are on $A B$ and $A C$ , respectively, such that $X Y$ is parallel to $B C$ . Which of the two following equalities always hold? (Here, $[P Q R]$ denotes the area of $Δ P Q R$ ).

$I$ . $[B C X] = [B C Y]$

$II$ . $[A C X] \cdot [A B Y] = [A X Y] \cdot [A B C]$

MEDIUM

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

In a

∆ A B C,

a = 2 x, b = 2 y

and

∠ C = 120 °,

then the area of the triangle is

EASY

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

If the sides of triangle are

4, 5

and

6 cm

.

Then the area (in sq cm) of triangle is

MEDIUM

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

Let

X, Y, Z

be respectively the areas of a regular pentagon, regular hexagon and regular heptagon which are inscribed in a circle of radius

1

. Then

HARD

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

In a quadrilateral

A B C D,

it is given that

A B = A D = 13, B C = C D = 20, B D = 24 .

r

is the radius of the circle inscribable in the quadrilateral, then what is the integer closest to

r ?

HARD

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

In the figure given below, $A B C D E F$ is a regular hexagon of side length $1$ unit, $A F P S$ and $A B Q R$ are squares. Then the ratio $\frac{area of △ A P Q}{area of △ S R P}$ equals

Question Image

HARD

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

15

and

9

are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer?

MEDIUM

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

With usual notations, in

∆ A B C

, if

b \cos^{2} \frac{C}{2} + c \cos^{2} \frac{B}{2} = \frac{3 a}{2}

, then

MEDIUM

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

Let

A B C D

be a convex cyclic quadrilateral. Suppose

P

is a point in the plane of the quadrilateral such that sum of its distances from the vertices of

A B C D

is the least. If

{P A, P B, P C, P D} = \{3, 4, 6, 8\}

, what is the maximum possible area of

A B C D ?

MEDIUM

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

a, b, c

are the sides of a

∆ A B C

and exradii

r_{1}, r_{2}, r_{3}

are respectively

12, 6, 4

then

a + 2 b + 3 c =

MEDIUM

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

A triangle

A B C

has area of

P

square units and circumference

2 S

units. If

h_{1}, h_{2}

and

h_{3}

are respectively the length of the altitudes of the triangle drawn from the vertices

A, B

and

C,

then

P^{2} [\frac{{(h_{1} h_{2} + h_{2} h_{3} + h_{3} h_{1})}^{2}}{h_{1}^{2} h_{2}^{2} h_{3}^{2}} - 2] =

MEDIUM

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

Consider four triangles having sides

(5, 12, 9), (5, 12, 11), (5, 12, 13)

and

(5, 12, 15)

. Among these the triangle having maximum area has sides

HARD

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

In a rectangle

A B C D,

points

X

and

Y

are the mid-points of

A D

and

D C

respectively. Lines

B X

and

C D

when extended intersect at

E

and lines

B Y

and

A D

when extended intersect at

F

. If the area of rectangle

A B C D

60

square units, then the area of

△ B E F

(in square units) is

MEDIUM

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

In a

∆ A B C

, if

\tan \frac{A}{2} = \frac{5}{6}, \tan \frac{C}{2} = \frac{2}{5}

, then

Suppose we have two circles of radius 2 each in the plane such that the distance between their centres is 23. The area of the region common to both circles lies between

Important Questions on Properties of Triangle

Learn and Prepare for any exam you want !

Suppose we have two circles of radius $2$ each in the plane such that the distance between their centres is $2 \sqrt{3}$ . The area of the region common to both circles lies between