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

Mention the formulae to find the in-radius (Apothem) of a regular polygon, when its (a) side length is given (b) circum-radius is given

Important Questions on Properties of Triangle

HARD

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

In an equilateral triangle

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

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>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 Triangle>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 Triangle>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 Triangle>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

Mention the formulae to find the in-radius (Apothem) of a regular polygon, when its (a) side length is given (b) circum-radius is given

Important Questions on Properties of Triangle

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