ln a △ A B C , A = 30 ° + C and R = ( 3 + 1 ) r , where r is the inradius and R is the circumradius, then

A B C is a right-angled triangle

A B C is an equilateral triangle

∠ A = 75 ° , ∠ B = 60 ° , ∠ C = 45 °

In a △ X Y Z let x , y , z be the lengths of sides opposite to the angles X , Y , Z respectively and 2 s = x + y + z . If s - x 4 = s - y 3 = s - z 2 and area of incircle of the triangle △ X Y Z is 8 π 3 , then

The radius of circumcircle of the triangle △ X Y Z is 35 6 6

EASY

Earn 100

In $∆ A B C,$ the circle that touches the sides $B C$ internally and other two sides $A B$ and $A C$ externally, is called

36.31% studentsanswered this correctly

Important Questions on Properties of Triangles

HARD

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

In a triangle

A B C

, a point

D

is chosen on

B C

such that

B D : D C = 2 : 5

. Let

P

be a point on the circumcircle

A B C

such that

∠ P D B = ∠ B A C

. Then

P D : P C

is :-

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

Δ A B C, \frac{r_{1} - r}{a} + \frac{r_{2} - r}{b} + \frac{r_{3} - r}{c} =

HARD

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

p_{1}, p_{2}, p_{3}

are the altitudes of a triangle

A B C

from the vertices

A, B, C

respectively, then with the usual notation,

\frac{1}{r_{1}^{2}} + \frac{1}{r_{2}^{2}} + \frac{1}{r_{3}^{2}} + \frac{1}{r^{2}} =

HARD

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

Let $A B C$ be a triangle with $∠ B A C = 90 °$ and $D$ be the point on the side $B C$ such that $A D ⊥ B C .$ Let $r, r_{1}$ and $r_{2}$ be the inradius of triangles $A B C, A B D,$ and $A C D$ , respectively. If $r, r_{1},$ and $r_{2}$ are positive integers and one of them is $5$ find the largest possible value of $r + r_{1} + r_{2}$ .

HARD

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

If the lengths of the sides of a triangle are

15, 20, 25

units. Find the circumradius of the triangle.

HARD

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

In a triangle

A B C,

the median from

B

C A

is perpendicular to the median from

C

A B

. If the median from

A

B C

30,

determine

(\frac{B C^{2} + C A^{2} + A B^{2}}{100})

EASY

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

The angles of a triangle are in the ratio

2 : 3 : 7

and the radius of the circumscribed circle is

10 cm .

The length of the smallest side is

EASY

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

In a triangle

A B C,

with usual notation, if

a = 12, b = 16, c = 20,

then the ratio of the exradii of the triangle opposite to the angles in the order

∠ C, ∠ B, ∠ A

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

ln a

△ A B C

A = 30 ° + C

and

R = (\sqrt{3} + 1) r

, where

r

is the inradius and

R

is the circumradius, then

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

d_{1}, d_{2}, d_{3}

are the diameters of three ex-circles of a

Δ A B C

, then

d_{1} d_{2} + d_{2} d_{3} + d_{3} d_{1} =

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

In a triangle

(1 - \frac{r_{1}}{r_{2}}) (1 - \frac{r_{1}}{r_{3}}) = 2,

then the triangle is

HARD

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

Three circles of radii

1, 2

and

3

units respectively touch each other externally in the plane. The circumradius of the triangle formed by joining the centres of the circles is

HARD

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

∆ A B C

, suppose the radius of the circle opposite to an angle

A

is denoted by

r_{1}

, similarly

r_{2} \leftrightarrow

angle

B, r_{3} \leftrightarrow

angle

C .

' r'

is the radius of inscribed circle then, what is the value of

\frac{a b - r_{1} r_{2}}{r_{3}} =

HARD

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

In a triangle

A B C,

right-angled at

A,

the altitude through

A

and the internal bisector of

∠ A

have lengths

3

units and

4

units, respectively. Find the length of the medium through

A .

EASY

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

In a

∆ A B C,

(a - b) (s - c) = (b - c) (s - a)

then

r_{1}, r_{2}

and

r_{3}

are

EASY

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

In an equilateral triangle

A B C, r : R : r_{1}

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

In a triangle

A B C,

a : b : c = 4 : 5 : 6,

then

\frac{1}{4 R} [r_{1} + r_{2} + r_{3}] =

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

Δ A B C

, if

a : b : c = 4 : 5 : 6

, then the ratio of the circumradius to its inradius is

HARD

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

Let

X Y

be the diameter of a semicircle with center

O .

Let

A

be a variable point on the semicircle and

B

another point on the semicircle such that

A B

is parallel to

X Y .

The value of

∠ B O Y

for which the inradius of triangle

A O B

is maximum, is

HARD

Mathematics>Trigonometry>Properties of Triangles>Inradius, Exradii and Circumradius

In a

△ X Y Z

let

x, y, z

be the lengths of sides opposite to the angles

X, Y, Z

respectively and

2 s = x + y + z .

\frac{s - x}{4} = \frac{s - y}{3} = \frac{s - z}{2}

and area of incircle of the triangle

△ X Y Z

\frac{8 π}{3}

, then

In ∆ABC, the circle that touches the sides BC internally and other two sides AB and AC externally, is called

Important Questions on Properties of Triangles

Learn and Prepare for any exam you want !

In $∆ A B C,$ the circle that touches the sides $B C$ internally and other two sides $A B$ and $A C$ externally, is called