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 = 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

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The point where the $3$ altitudes meet is called the ortho-centre of the triangle.

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Important Questions on Properties of Triangle

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

If the lengths of the sides of a triangle are

15, 20, 25

units. Find the circumradius of the triangle.

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

A line cuts the

x

-axis at

A (7, 0)

and the

y

-axis at

B (0, - 5)

. A variable line

P Q

is drawn perpendicular to

A B

cutting the

x

-axis at

P (a, 0)

and the

y

-axis at

Q (0, b)

. If

A Q

and

B P

intersect at

R

, the locus of

R

MEDIUM

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

Construct an incircle of an equilateral triangle with side

5 cm

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

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

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})

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

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

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}} =

MEDIUM

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

$ABC$ is an equilateral triangle of side $2 a$ , then length of one of its altitude is $a \sqrt{x}$ . The value of $a$ is _____.

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

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 .

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

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

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

MEDIUM

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

Construct a cyclic quadrilateral

A B C D

in which

A B = 6 cm, A C = 7 cm, B C = 6 cm

and

A D = 4.2 cm

EASY

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

∆ A B C, A B = A C

and

A L

is perpendicular to

B C

L

. In

∆ D E F, D E = D F

and

D M

is perpendicular to

E F

M

. If

(a r e a o f ∆ A B C) : (a r e a o f ∆ D E F) = 9.25

, then

\frac{D M + A L}{D M - A L}

is equal to:

The point where the 3 altitudes meet is called the ortho-centre of the triangle.

Important Questions on Properties of Triangle

Learn and Prepare for any exam you want !

The point where the $3$ altitudes meet is called the ortho-centre of the triangle.