# JEE Main > Properties of Triangle

Explore popular questions from Properties of Triangle for JEE Main. This collection covers Properties of Triangle previous year JEE Main questions hand picked by popular teachers.

Physics
Chemistry
Maths

Q 1.

Correct4

Incorrect-1

Three distinct points {tex} A , B {/tex} and {tex} C {/tex} are given in the two-dimensional coordinate plane such that the ratio of the distance of any one of them from the point {tex} ( 1,0 ) {/tex} to the distance from the point {tex} ( - 1,0 ) {/tex} is equal to {tex} \frac { 1 } { 3 } {/tex} . Then the circumcentre of the triangle {tex} A B C {/tex} is at the point

A

{tex} \left( \frac { 5 } { 4 } , 0 \right) {/tex}

B

{tex} \left( \frac { 5 } { 2 } , 0 \right) {/tex}

C

{tex} \left( \frac { 5 } { 3 } , 0 \right) {/tex}

{tex} \left( \frac { 5 } { 4} , 0 \right) {/tex}

##### Explanation

Q 2.

Correct4

Incorrect-1

For a regular polygon, let' r' and 'R' be the radii of the inscribed and the circumscribed circles. A false statement among the following is

A

There is a regular polygon with {tex} \frac { r } { R } = \frac { 1 } { \sqrt { 2 } } {/tex}

There is a regular polygon with {tex} \frac { r } { R } = \frac { 2 } { 3 } {/tex}

C

There is a regular polygon with {tex} \frac { r } { R } = \frac { \sqrt { 3 } } { 2 } {/tex}

D

There is a regular polygon with {tex} \frac { r } { R } = \frac { 1 } { 2 } {/tex}

##### Explanation

Q 3.

Correct4

Incorrect-1

{tex} A B C D {/tex} is a trapezium such that {tex} A B {/tex} and {tex} C D {/tex} are parallel and {tex} B C \perp C D . {/tex} If {tex} \angle A D B = \theta , B C = p {/tex} and {tex} C D = q {/tex} , then {tex} A B {/tex} is equal to

A

{tex} \frac { p ^ { 2 } + q ^ { 2 } \cos \theta } { p \cos \theta + q \sin \theta } {/tex}

B

{tex} \frac { p ^ { 2 } + q ^ { 2 } } { p ^ { 2 } \cos \theta + q ^ { 2 } \sin \theta } {/tex}

C

{tex} \frac { \left( p ^ { 2 } + q ^ { 2 } \right) \sin \theta } { ( p \cos \theta + q \sin \theta ) ^ { 2 } } {/tex}

{tex} \frac { \left( p ^ { 2 } + q ^ { 2 } \right) \sin \theta } { p \cos \theta + q \sin \theta } {/tex}

##### Explanation

Q 4.

Correct4

Incorrect-1

The angle of elevation of the top of a vertical tower from a point {tex} P {/tex} on the horizontal ground was observed to be {tex} \alpha . {/tex} After moving a distance 2{tex} \mathrm { m } {/tex} from {tex} P {/tex} towards the foot of the tower, the angle of elevation changes to {tex} \beta . {/tex} Then the height (in metres) of the tower is

{tex} \frac { 2 \sin \alpha \sin \beta } { \sin ( \beta - \alpha ) } {/tex}

B

{tex} \frac { \sin \alpha \sin \beta } { \cos ( \beta - \alpha ) } {/tex}

C

{tex} \frac { 2 \sin ( \beta - \alpha ) } { \sin \alpha \sin \beta } {/tex}

D

{tex} \frac { \cos ( \beta - \alpha ) } { \sin \alpha \sin \beta } {/tex}

##### Explanation

Q 5.

Correct4

Incorrect-1

From the top of a 64{tex} \mathrm { m } {/tex} high tower, a stone is thrown upwards vertically with the velocity of 48{tex} \mathrm { m } / \mathrm { s } {/tex} . The greatest height (in meters) attained by the stone, assuming the value of the gravi- tational acceleration, {tex} g = 32 \mathrm { m } / \mathrm { s } ^ { 2 } {/tex} is

100

B

88

C

128

D

112

##### Explanation

Q 6.

Correct4

Incorrect-1

{tex} A B C {/tex} is a triangle in a plane with vertices {tex} A ( 2,3,5 ) , B ( - 1,3,2 ) {/tex} and {tex} C ( \lambda , 5 , \mu ) {/tex} . If the median through {tex} A {/tex} is equally inclined to the coordinate axes, then the value of {tex} \left( \lambda ^ { 3 } + \mu ^ { 3 } + 5 \right) {/tex} is

A

1130

1348

C

1077

D

676

##### Explanation

Q 7.

Correct4

Incorrect-1

If the lengths of arcs {tex} A B , B C {/tex} and {tex} C A {/tex} of a circle are {tex} 3,4 {/tex} and {tex} 5 , {/tex} respectively, then the area of triangle {tex} A B C {/tex} is

{tex} \frac { 9 \sqrt { 3 } ( \sqrt { 3 } + 1 ) } { \pi ^ { 2 } } {/tex}

B

{tex} \frac { 9 \sqrt { 3 } ( \sqrt { 3 } - 1 ) } { \pi ^ { 2 } } {/tex}

C

{tex} \frac { 9 \sqrt { 3 } ( \sqrt { 3 } - 1 ) } { \pi } {/tex}

D

None of these

##### Explanation

Q 8.

Correct4

Incorrect-1

The area of right-angled triangle in terms of {tex} r {/tex} and {tex} r _ { 1 } {/tex} , if {tex} \angle A = 90 ^ { \circ } {/tex} (where {tex} r , r _ { 1 } {/tex} have their usual meanings), is

A

{tex} r + r _ { 1 } {/tex}

{tex} rr_1 {/tex}

C

{tex} r - r _ { 1 } {/tex}

D

{tex} r _ { 1 } - r {/tex}

##### Explanation

Q 9.

Correct4

Incorrect-1

If in a {tex} \Delta A B C {/tex} (whose circumcentre is origin), {tex} a \leq \sin A , {/tex} then for any point {tex} ( x , y ) {/tex} inside the circumcircle of {tex} \Delta A B C {/tex}

{tex} | x y | < 1 / 8 {/tex}

B

{tex} | x y | > 1 / 8 {/tex}

C

{tex} 1 / 8 < x y < 1 / 2 {/tex}

D

None of these

##### Explanation

Q 10.

Correct4

Incorrect-1

If the sine of the angles of a triangle {tex} A B C {/tex} satisfy the equation {tex} c ^ { 3 } x ^ { 3 } - c ^ { 2 } ( a + b + c ) x ^ { 2 } + \lambda x + \mu = 0 {/tex} (where {tex} a , b , c {/tex} are the sides of {tex} \Delta A B C {/tex} , then triangle {tex} A B C {/tex} is

A

always right-angled for any {tex} \lambda , \mu {/tex}

right-angled only when {tex} \lambda = c ( a b + b c + c a ) , \mu = - a b c {/tex}

C

right-angled only when {tex} \lambda = \frac { c ( a b + b c + c a ) } { 4 } , \mu = \frac { - a b c } { 8 } {/tex}

D

never right-angled

##### Explanation

Q 11.

Correct4

Incorrect-1

If {tex} \sin A {/tex} and {tex} \sin B {/tex} of a triangle {tex} A B C {/tex} satisfy {tex} c ^ { 2 } x ^ { 2 } - c ( a + b ) x + a b = 0 {/tex} then the triangle is

A

equilateral

B

isosceles

right-angled

D

acute angled

##### Explanation

Q 12.

Correct4

Incorrect-1

{tex} A B C D {/tex} is a quadrilateral circumscribed about a circle of unit radius. Then

A

{tex} A B \sin \frac { C } { 2 } \cdot \sin \frac { A } { 2 } = C D \sin \frac { B } { 2 } \cdot \sin \frac { D } { 2 } {/tex}

{tex} A B \sin \frac { A } { 2 } \cdot \sin = \frac { B } { 2 } = C D \sin \frac { C } { 2 } \cdot \sin \frac { D } { 2 } {/tex}

C

{tex} A B \sin \frac { A } { 2 } \cdot \sin \frac { D } { 2 } = C D \sin \frac { C } { 2 } \cdot \sin \frac { B } { 2 } {/tex}

D

{tex} A B \sin \frac { A } { 2 } \cdot \cos = \frac { B } { 2 } = C D \sin \frac { C } { 2 } \cdot \cos \frac { D } { 2 } {/tex}

##### Explanation

Q 13.

Correct4

Incorrect-1

If in triangle {tex} A B C {/tex} , line joining the circumcentre and orthocentre is parallel to side {tex} A C , {/tex} then value of {tex} \tan A\cdot{/tex}tan {tex} C {/tex} is equal to

A

{tex} \sqrt { 3 } {/tex}

3

C

3{tex} \sqrt { 3 } {/tex}

D

None of these

##### Explanation

Q 14.

Correct4

Incorrect-1

{tex} A A _ { 1 } , B B _ { 1 }, C C _ { 1 } {/tex} are the medians of triangle {tex} A B C {/tex} whose centroid is {tex} G . {/tex} If the points {tex} A , C _ { 1 } {/tex} and {tex} B _ { 1 } {/tex} are concyclic, then

A

{tex} 2 b ^ { 2 } = a ^ { 2 } + c ^ { 2 } {/tex}

B

{tex} 2 c ^ { 2 } = a ^ { 2 } + b ^ { 2 } {/tex}

{tex} 2 a ^ { 2 } = b ^ { 2 } + c ^ { 2 } {/tex}

D

None of these

##### Explanation

Q 15.

Correct4

Incorrect-1

The area of a triangle {tex} A B C , {/tex} where {tex} a = 2 ( \sqrt { 3 } + 1 ) , B = 45 ^ { \circ } , C = 60 ^ { \circ } {/tex} is

A

{tex} \sqrt { 3 } ( \sqrt { 3 } + 1 ) {/tex} square unit

B

2{tex} ( \sqrt { 3 } + 1 ) {/tex} square unit

2{tex} \sqrt { 3 } ( \sqrt { 3 } + 1 ) {/tex} square unit

D

{tex} \sqrt { 3 } ( 2 \sqrt { 3 } + 1 ) {/tex} square unit

##### Explanation

Q 16.

Correct4

Incorrect-1

In a triangle {tex} A B C {/tex} , the value of $\frac { \cos ^ { 2 } B - \cos ^ { 2 } C } { b + c } + \frac { \cos ^ { 2 } C - \cos ^ { 2 } A } { c + a } + \frac { \cos ^ { 2 } A - \cos ^ { 2 } B } { a + b } \text { is }$

0

B

1

C

2

D

3

##### Explanation

Q 17.

Correct4

Incorrect-1

In a triangle {tex} A B C {/tex} if {tex} \cos A + 2 \cos B + \cos C = 2 , {/tex} the sides of the triangle are in

A

{tex} \mathrm { HP } {/tex}

B

{tex} G P {/tex}

{tex} A P {/tex}

D

None of these

##### Explanation

Q 18.

Correct4

Incorrect-1

In a triangle, {tex} 1 - \tan \frac { A } { 2 } \tan \frac { B } { 2 } = {/tex}

A

{tex} \frac { 2 } { a + b + c } {/tex}

{tex} \frac { 2 c } { a + b + c } {/tex}

C

{tex} \frac { c } { a + b + c } {/tex}

D

None of these

##### Explanation

Q 19.

Correct4

Incorrect-1

In a triangle ABC
{tex} a ^ { 2 } b ^ { 2 } c ^ { 2 } ( \sin 2 A + \sin 2 B + \sin 2 C ) = {/tex}

A

{tex} \Delta ^ { 3 } {/tex}

B

8{tex} \Delta ^ { 3 } {/tex}

C

16{tex} \Delta ^ { 3 } {/tex}

32{tex} \Delta ^ { 3 } {/tex}

##### Explanation

Q 20.

Correct4

Incorrect-1

If ex-radii {tex} r _ { 1 } , r _ { 2 } , r _ { 3 } {/tex} of a triangle are in HP then its sides {tex} a , b , c {/tex} are in

{tex} A P {/tex}

B

{tex} G P {/tex}

C

{tex} \mathrm { HP } {/tex}

D

None of these

##### Explanation

Q 21.

Correct4

Incorrect-1

In a right-angled {tex} \Delta A B C , \sin ^ { 2 } A + \sin ^ { 2 } B + \sin ^ { 2 } C {/tex} is equal to

A

0

B

1

C

-1

None of these

##### Explanation

Q 22.

Correct4

Incorrect-1

In a right-angled {tex} \triangle A B C , \sin ^ { 2 } A + \sin ^ { 2 } B + \sin ^ { 2 } C {/tex} is equal to

A

{tex}\Delta{/tex}

B

2{tex}\Delta{/tex}

C

3{tex} \Delta {/tex}

4{tex} \Delta {/tex}

##### Explanation

Q 23.

Correct4

Incorrect-1

If {tex} p _ { 1 } , p _ { 2 } , p _ { 3 } {/tex} are, respectively, the perpendiculars from the vertices of a triangle to the opposite sides, then {tex} p _ { 1 } p _ { 2 } p _ { 3 } {/tex} is equal to

A

{tex} \frac { a ^ { 2 } b ^ { 2 } c ^ { 2 } } { R ^ { 2 } } {/tex}

B

{tex} \frac { a ^ { 2 } b ^ { 2 } c ^ { 2 } } { 4 R ^ { 2 } } {/tex}

C

{tex} \frac { 4 a ^ { 2 } b ^ { 2 } c ^ { 2 } } { R ^ { 2 } } {/tex}

{tex} \frac { a ^ { 2 } b ^ { 2 } c ^ { 2 } } { 8 R ^ { 2 } } {/tex}

##### Explanation

Q 24.

Correct4

Incorrect-1

If {tex} p _ { 1 } , p _ { 2 } , p _ { 3 } {/tex} are, respectively, the perpendiculars from the vertices of a triangle to the opposite sides, then {tex} \frac { \cos A } { p _ { 1 } } + \frac { \cos B } { p _ { 2 } } + \frac { \cos C } { p _ { 3 } } {/tex} is equal to

A

1{tex} / r {/tex}

1{tex} / R {/tex}

C

1{tex} / \Delta {/tex}

D

None of these

##### Explanation

Q 25.

Correct4

Incorrect-1

If {tex} \Delta = a ^ { 2 } - ( b - c ) ^ { 2 } , {/tex} where {tex} \Delta {/tex} is the area of triangle {tex} A B C , {/tex} then {tex} \tan A {/tex} is equal to

A

{tex} \frac { 15 } { 16 } {/tex}

{tex} \frac { 8 } { 15 } {/tex}

C

{tex} \frac { 8 } { 17 } {/tex}

D

{tex} \frac { 1 } { 2 } {/tex}