Properties of Triangles

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JEE Main 2013

$A B C D$ is a trapezium such that $A B$ and $C D$ are parallel and $B C ⊥ C D$ . If $∠ A D B = θ, B C = p$ and $C D = q$ , then $A B$ is equal to

\frac{( p ^{2} + q ^{2} ) s i n θ}{p c o s θ + q s i n θ}

(B)

\frac{p ^{2} + q ^{2} c o s θ}{p c o s θ + q s i n θ}

(C)

\frac{p ^{2} + q ^{2}}{p ^{2} c o s θ + q ^{2} s i n θ}

(D)

\frac{( p ^{2} + q ^{2} ) s i n θ}{( p c o s θ + q s i n θ ) ^{2}}

JEE Main 2010

LEVELJEE Main

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

\frac{r}{R} = \frac{1}{2}

(B)

There is a regular polygon with

\frac{r}{R} = \frac{2}{3}

(C)

There is a regular polygon with

\frac{r}{R} = \frac{3}{2}

(D)

There is a regular polygon with

\frac{r}{R} = \frac{1}{2}

JEE Main 2005

LEVELJEE Main

In a triangle $A B C$ , let $∠ C = \frac{π}{2}$ . If $r$ is the inradius and $R$ is the circumradius of the triangle $A B C$ , then $2 (r + R)$ equals

(A)

b + c

(B)

a + b

(C)

a + b + c

(D)

c + a

JEE Main 2005

LEVELJEE Main

If in a $Δ A B C$ , the altitudes from the vertices $A, B, C$ on opposite sides are in H.P, then $sin A, sin B, sin C$ are in

Three forces $P, Q$ and $R$ acting along IA, IB and IC, where I is the incentre of a $△ A B C$ are in equilibrium. Then $P : Q : R$ is

(A)

csc \frac{A}{2} : csc \frac{B}{2} : csc \frac{C}{2}

(B)

sin \frac{A}{2} : sin \frac{B}{2} : sin \frac{C}{2}

(C)

sec \frac{A}{2} : sec \frac{B}{2} : sec \frac{C}{2}

(D)

cos \frac{A}{2} : cos \frac{B}{2} : cos \frac{C}{2}

JEE Main 2004

LEVELJEE Main

The sides of a triangle are $sin α, cos α$ and $1 + sin α cos α$ for some $0 < α < \frac{π}{2}$ . Then the greatest angle of the triangle is

(A)

15 0^{\circ}

(B)

9 0^{\circ}

(C)

12 0^{\circ}

(D)

6 0^{\circ}

JEE Main 2003

LEVELJEE Main

The sum of the radii of inscribed and circumscribed circles for an $n$ sided regular polygon of side $a$ , is

(A)

\frac{a}{4} cot (\frac{π}{2 n})

(B)

a cot (\frac{π}{n})

(C)

\frac{a}{2} cot (\frac{π}{2 n})

(D)

a cot (\frac{2 π}{n})

JEE Main 2003

LEVELJEE Main

In a triangle $A B C$ , medians $A D$ and $B E$ are drawn. If $A D = 4$ , $∠ D A B = \frac{π}{6}$ and $∠ A B E = \frac{π}{3}$ , then the area of the $Δ A B C$ is

(A)

\frac{64}{3}

(B)

\frac{8}{3}

(C)

\frac{16}{3}

(D)

\frac{32}{3 3}

JEE Main 2003

LEVELJEE Main

If in a $Δ A B C$ $a cos^{2} (\frac{C}{2}) + c cos^{2} (\frac{A}{2}) = \frac{3 b}{2}$ , then the sides $a, b$ and $c$

(A)

satisfy

a + b = c

The sides of a triangle are $3 x + 4 y$ , $4 x + 3 y$ and $5 x + 5 y$ where $x, y > 0$ then the triangle is

In a triangle with sides $a, b, c, r_{1} > r_{2} > r_{3}$ (which are the ex-radii) then

(A)

a > b > c

(B)

a < b < c

(C)

a > b

and

b < c

(D)

a < b

and

b > c

JEE Advanced 2016

LEVELJEE Advanced

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

* Multiple Correct Options

(A)

area of the triangle

X Y Z

66

(B)

the radius of circumcircle of the triangle

X Y Z

\frac{35}{6} 6

(C)

sin \frac{X}{2} sin \frac{Y}{2} sin \frac{Z}{2} = \frac{4}{35}

(D)

sin^{2} (\frac{X + Y}{2}) = \frac{3}{5}

JEE Advanced 2014

LEVELJEE Advanced

In a triangle the sum of two sides is $x$ and the product of the same sides is $y$ . If $x^{2} - c^{2} = y$ , where $c$ is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is

(A)

\frac{3 y}{2 x ( x + c )}

(B)

\frac{3 y}{2 c ( x + c )}

(C)

\frac{3 y}{4 x ( x + c )}

(D)

\frac{3 y}{4 c ( x + c )}

JEE Advanced 2013

LEVELJEE Main

In a triangle $P QR, P$ is the largest angle and $cos P = 1/3$ . Further the incircle of the triangle touches the sides $P Q, QR$ and $R P$ at $N, L$ and $M$ respectively, such that the lengths of $P N, Q L$ and $R M$ are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

* Multiple Correct Options

Let $P QR$ be a triangle of area $Δ$ with $a = 2, b = 7/2$ and $c = 5/2$ , where $a, b$ , and $c$ are the lengths of the sides of the triangle opposite to the angles at $P, Q$ and $R$ respectively. Then $\frac{2 s i n P - s i n 2 P}{2 s i n P + s i n 2 P}$ equals

(A)

\frac{3}{4Δ}

(B)

\frac{45}{4Δ}

(C)

(\frac{3}{4Δ})^{2}

(D)

(\frac{45}{4Δ})^{2}

JEE Advanced 2010

LEVELJEE Main

If the angles $A, B$ and $C$ of a triangle are in an arithmetic progression and if $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively, then the value of the expression $\frac{a}{c} sin 2 C + \frac{c}{a} sin 2 A$ is

(A)

1/2

(B)

3 /2

(C)

(D)

3

JEE Advanced 2010

LEVELJEE Main

Let $A B C$ be a triangle such that $∠ A C B = π /6$ and let $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively. The value(s) of $x$ for which $a = x^{2} + x + 1, b = x^{2} - 1$ and $c = 2 x + 1$ is (are)

(A)

- (2 + 3)

(B)

1 + 3

(C)

2 + 3

(D)

4 e x t \sqrt 3

JEE Advanced 2009

LEVELJEE Advanced

In a triangle $A B C$ with fixed base $B C$ , the vertex $A$ moves such that $cos B + cos C = 4 sin^{2} \frac{A}{2}$ . If $a, b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B$ and $C$ , respectively, then

* Multiple Correct Options

(A)

b + c = 4 a

(B)

b + c = 2 a

(C)

locus of point

A

is an ellipse

(D)

locus of point

A

is a pair of straight lines

JEE Advanced 2006

LEVELJEE Main

One angle of an isosceles $Δ$ is $12 0^{\circ}$ and radius of its incircle $= 3$ . Then the area of the triangle in sq. units is

(A)

7 + 123

(B)

12 - 73

(C)

12 + 73

(D)

4 π

JEE Advanced 2006

LEVELJEE Main

In $Δ A B C$ , internal angle bisector of $∠ A$ meets side $B C$ in $D$ . $D E ⊥ A D$ meets $A C$ in $E$ and $A B$ in $F$ . Then

* Multiple Correct Options

(A)

A E

is HM of

b

c

(B)

A D = \frac{2 b c}{b + c} cos \frac{A}{2}

(C)

E F = \frac{4 b c}{b + c} sin \frac{A}{2}

(D)

Δ A E F

is isosceles

JEE Advanced 2005

LEVELJEE Main

In an equilateral triangle, 3 coins of radii 1 unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is

(A)

4 + 23

(B)

6 + 43

(C)

12 + \frac{7 3}{4}

(D)

3 + \frac{7 3}{4}

JEE Advanced 2004

LEVELJEE Main

The sides of a triangle are in the ratio $1 : 3 : 2$ , then the angles of the triangle are in the ratio

(A)

1 : 3 : 5

(B)

2 : 3 : 4

(C)

3 : 2 : 1

(D)

1 : 2 : 3

JEE Advanced 2003

LEVELBoard

If the angles of a triangle are in the ratio $4 : 1 : 1$ , then the ratio of the longest side to the perimeter is

(A)

3 : (2 + 3)

(B)

1 : 6

(C)

1 : 2 + 3

(D)

2 : 3

JEE Advanced 2002

LEVELJEE Main

Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $A B C$ ( $R$ being the radius of the circumcircle)?

(A)

a, sin A, sin B

(B)

a, b, c

(C)

a, sin B, R

(D)

a, sin A, R

JEE Advanced 2000

LEVELBoard

In a triangle $A B C$ , $2 a c sin \frac{1}{2} (A - B + C) =$

(A)

a^{2} + b^{2} - c^{2}

(B)

c^{2} + a^{2} - b^{2}

(C)

b^{2} - c^{2} - a^{2}

(D)

c^{2} - a^{2} - b^{2}

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ABCD is a trapezium such that AB and CD are parallel and BC⊥CD. If ∠ADB=θ,BC=p and CD=q, then AB is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a triangle ABC, let ∠C=2π​. If r is the inradius and R is the circumradius of the triangle ABC, then 2(r+R) equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If in a ΔABC, the altitudes from the vertices A,B,C on opposite sides are in H.P, then sinA,sinB,sinC are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Three forces P,Q​ and R acting along IA, IB and IC, where I is the incentre of a △ABC are in equilibrium. Then P:Q:R is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sides of a triangle are sinα,cosα and 1+sinαcosα​ for some 0<α<2π​. Then the greatest angle of the triangle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a triangle ABC, medians AD and BE are drawn. If AD=4, ∠DAB=6π​ and ∠ABE=3π​, then the area of the ΔABC is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If in a ΔABC acos2(2C​)+ccos2(2A​)=23b​, then the sides a,b and c

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sides of a triangle are 3x+4y, 4x+3y and 5x+5y where x,y>0 then the triangle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a triangle with sides a,b,c,r1​>r2​>r3​ (which are the ex-radii) then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } One angle of an isosceles Δ is 120∘ and radius of its incircle =3​. Then the area of the triangle in sq. units is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In ΔABC, internal angle bisector of ∠A meets side BC in D. DE⊥AD meets AC in E and AB in F. Then

In an equilateral triangle, 3 coins of radii 1 unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sides of a triangle are in the ratio 1:3​:2, then the angles of the triangle are in the ratio

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the angles of a triangle are in the ratio 4:1:1, then the ratio of the longest side to the perimeter is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Which of the following pieces of data does NOT uniquely determine an acute-angled triangle ABC (R being the radius of the circumcircle)?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a triangle ABC, 2acsin21​(A−B+C)=

$A B C D$ is a trapezium such that $A B$ and $C D$ are parallel and $B C ⊥ C D$ . If $∠ A D B = θ, B C = p$ and $C D = q$ , then $A B$ is equal to

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

In a triangle $A B C$ , let $∠ C = \frac{π}{2}$ . If $r$ is the inradius and $R$ is the circumradius of the triangle $A B C$ , then $2 (r + R)$ equals

If in a $Δ A B C$ , the altitudes from the vertices $A, B, C$ on opposite sides are in H.P, then $sin A, sin B, sin C$ are in

Three forces $P, Q$ and $R$ acting along IA, IB and IC, where I is the incentre of a $△ A B C$ are in equilibrium. Then $P : Q : R$ is

The sides of a triangle are $sin α, cos α$ and $1 + sin α cos α$ for some $0 < α < \frac{π}{2}$ . Then the greatest angle of the triangle is

The sum of the radii of inscribed and circumscribed circles for an $n$ sided regular polygon of side $a$ , is

In a triangle $A B C$ , medians $A D$ and $B E$ are drawn. If $A D = 4$ , $∠ D A B = \frac{π}{6}$ and $∠ A B E = \frac{π}{3}$ , then the area of the $Δ A B C$ is

If in a $Δ A B C$ $a cos^{2} (\frac{C}{2}) + c cos^{2} (\frac{A}{2}) = \frac{3 b}{2}$ , then the sides $a, b$ and $c$

The sides of a triangle are $3 x + 4 y$ , $4 x + 3 y$ and $5 x + 5 y$ where $x, y > 0$ then the triangle is

In a triangle with sides $a, b, c, r_{1} > r_{2} > r_{3}$ (which are the ex-radii) then

One angle of an isosceles $Δ$ is $12 0^{\circ}$ and radius of its incircle $= 3$ . Then the area of the triangle in sq. units is

In $Δ A B C$ , internal angle bisector of $∠ A$ meets side $B C$ in $D$ . $D E ⊥ A D$ meets $A C$ in $E$ and $A B$ in $F$ . Then

The sides of a triangle are in the ratio $1 : 3 : 2$ , then the angles of the triangle are in the ratio

If the angles of a triangle are in the ratio $4 : 1 : 1$ , then the ratio of the longest side to the perimeter is

Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $A B C$ ( $R$ being the radius of the circumcircle)?

In a triangle $A B C$ , $2 a c sin \frac{1}{2} (A - B + C) =$