In radius of a circle which is inscribed in a isosceles triangle one of whose angle is 23 is 3 then areaof triangle in sq units is

MEDIUM

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

Let

A B C

be a triangle such that

A B = 4, B C = 5

and

C A = 6

. Choose points

D, E, F

on

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}

is

HARD

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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

is

MEDIUM

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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

MEDIUM

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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] =

HARD

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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?

EASY

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

If the sides of triangle are

4, 5

and

6 cm

.

Then the area (in sq cm) of triangle is

HARD

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

Denote Area

(∆ X Y Z), P (∆ X Y Z)

and

|X Y|

by area of the triangle

∆ X Y Z

, perimeter of the triangle

∆ X Y Z

and length of the line segment

X Y

respectively.
Let

A B C D

be a convex quadrangle and the diagonals

A C

and

B D

intersect at

O

. Then

HARD

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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

MEDIUM

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

In a

∆ A B C,

if

a = 2 x, b = 2 y

and

∠ C = 120 °,

then the area of the triangle is

MEDIUM

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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) =

MEDIUM

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

If

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 =

HARD

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

In a triangle

A B C,

if

\cot \frac{A}{2} \cot \frac{B}{2} = K,

then all the possible values of

K

lies in

MEDIUM

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

In a

Δ A B C,

if

4 a = b + c,

then

\tan \frac{B}{2} \tan \frac{C}{2} =

HARD

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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

HARD

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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

is

60

square units, then the area of

△ B E F

(in square units) is

MEDIUM

Mathematics>Geometry - III>Similarity, Right Triangles, and Trigonometry>Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

In a

∆ A B C

, if

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

, then

In radius of a circle which is inscribed in a isosceles triangle one of whose angle is $2 π / 3,$ is $\sqrt{3},$ then area of triangle (in sq units) is

Important Questions on Similarity, Right Triangles, and Trigonometry

Learn and Prepare for any exam you want !

In radius of a circle which is inscribed in a isosceles triangle one of whose angle is 2π/3, is 3, then area of triangle (in sq units) is

Important Questions on Similarity, Right Triangles, and Trigonometry

Learn and Prepare for any exam you want !

In radius of a circle which is inscribed in a isosceles triangle one of whose angle is $2 π / 3,$ is $\sqrt{3},$ then area of triangle (in sq units) is