Properties of Triangle · Mathematics · JEE Advanced

Numerical

$$ \text { Let } a \text { be the area of the triangle } A B C \text {. Then the value of }(64 a)^2 \text { is } $$ :

JEE Advanced 2023 Paper 2 Online

In a triangle ABC, let AB = $$\sqrt {23} $$, BC = 3 and CA = 4. Then the value of $${{\cot A + \cot C} \over {\cot B}}$$ is _________.

JEE Advanced 2021 Paper 1 Online

In a triangle PQR, let a = QR, b = RP, and c = PQ. If |a| = 3, |b| = 4

and $${{a\,.(\,c - \,b)} \over {c\,.\,(a - \,b)}} = {{|a|} \over {|a| + |b|}}$$, then the value of |a $$ \times $$ b|² is ......

JEE Advanced 2020 Paper 1 Offline

Consider a triangle $$ABC$$ and let $$a, b$$ and $$c$$ denote the lengths of the sides opposit to vertices $$A, B$$ and $$C$$ respectively. Suppose $$a = 6,b = 10$$ and the area of the triangle is $$15\sqrt 3 $$, if $$\angle ACB$$ is obtuse and if $$r$$ denotes the radius of the incircle of the triangle, then r² is equal to :

IIT-JEE 2010 Paper 2 Offline

Let ABC and ABC' be two non-congruent triangles with sides AB = 4, AC = AC' = 2$$\sqrt2$$ and angle B = 30$$^\circ$$. The absolute value of the difference between the areas of these triangles is ___________.

IIT-JEE 2009 Paper 2 Offline

MCQ (More than One Correct Answer)

Consider a triangle PQR having sides of lengths p, q and r opposite to the angles P, Q and R, respectively. Then which of the following statements is (are) TRUE?

JEE Advanced 2021 Paper 2 Online

Let x, y and z be positive real numbers. Suppose x, y and z are the lengths of the sides of a triangle opposite to its angles X, Y, and Z, respectively. If

$$\tan {X \over 2} + \tan {Z \over 2} = {{2y} \over {x + y + z}}$$, then which of the following statements is/are TRUE?

JEE Advanced 2020 Paper 1 Offline

In a non-right-angled triangle $$\Delta $$PQR, let p, q, r denote the lengths of the sides opposite to the angles At P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If p = $${\sqrt 3 }$$, q = 1, and the radius of the circumcircle of the $$\Delta $$PQR equals 1, then which of the following options is/are correct?

JEE Advanced 2019 Paper 1 Offline

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

JEE Advanced 2016 Paper 1 Offline

In a triangle $$PQR$$, $$P$$ is the largest angle and $$\cos P = {1 \over 3}$$. Further the incircle of the triangle touches the sides $$PQ$$, $$QR$$ and $$RP$$ at $$N,L$$ and $$M$$ respectively, such that the lengths of $$PN, QL$$ and $$RM$$ are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

JEE Advanced 2013 Paper 2 Offline

In $$\Delta ABC$$, internal angle bisector of $$\angle A$$ meets side $$BC$$ in $$D$$. $$DE \bot AD$$ meets $$AC$$ in $$E$$ and $$AB$$ in $$F$$. Then

IIT-JEE 2006

In a triangle, the lengths of the two larger sides are $$10$$ and $$9$$, respectively. If the angles are in $$AP$$. Then the length of the third side can be

IIT-JEE 1987

There exists a triangle $$ABC$$ satisfying the conditions

IIT-JEE 1986

MCQ (Single Correct Answer)

Match the following :

	Column I		Column I
(A)	$\begin{array}{l}\text { In a triangle } \Delta X Y Z \text {, let } a, b \text { and } c \text { be the lengths of the sides } \\\text { opposite to the angles } X, Y \text { and } Z \text {, respectively. If } 2\left(a^2-b^2\right)=c^2 \\\text { and } \lambda=\frac{\sin (X-Y)}{\sin Z} \text {, then possible values of } n \text { for which } \cos (n \lambda) \\=0 \text { is (are) }\end{array}$	(P)	1
(B)	$\begin{array}{l}\text { In a triangle } \triangle X Y Z \text {, let } a, b \text { and } c \text { be the lengths of the sides } \\\text { opposite to the angles } X, Y \text { and } Z \text {, respectively. If } 1+\cos 2 X-2 \\\cos 2 Y=2 \sin X \sin Y \text {, then possible value(s) of } \frac{a}{b} \text { is (are) }\end{array}$	(Q)	2
(C)	$\begin{array}{l}\text { In } \mathbb{R}^2 \text {, let } \sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j} \text { and } \beta \hat{i}+(1-\beta) \hat{j} \text { be the position } \\\text { vectors of } X, Y \text { and } Z \text { with respect of the origin } \mathrm{O} \text {, respectively. If } \\\text { the distance of } \mathrm{Z} \text { from the bisector of the acute angle of } \overrightarrow{\mathrm{OX}} \text { with } \\\overrightarrow{\mathrm{OY}} \text { is } \frac{3}{\sqrt{2}} \text {, then possible value(s) of }\|\beta\| \text { is (are) }\end{array}$	(R)	3
(D)	$\begin{array}{l}\text { Suppose that } F(\alpha) \text { denotes the area of the region bounded by } \\x=0, x=2, y^2=4 x \text { and } y=\|\alpha x-1\|+\|\alpha x-2\|+\alpha x \text {, } \\\text { where, } \alpha \in\{0,1\} \text {. Then the value(s) of } F(\alpha)+\frac{8}{2} \sqrt{2} \text {, when } \alpha=0 \\\text { and } \alpha=1 \text {, is (are) }\end{array}$	(S)	5
		(T)	6

JEE Advanced 2015 Paper 1 Offline

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

JEE Advanced 2014 Paper 2 Offline

Let $$PQR$$ be a triangle of area $$\Delta $$ with $$a=2$$, $$b = {7 \over 2}$$ and $$c = {5 \over 2}$$; 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 $${{2\sin P - \sin 2P} \over {2\sin P + \sin 2P}}$$ equals.

IIT-JEE 2012 Paper 2 Offline

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 $${a \over c}\sin 2C + {c \over a}\sin 2A$$ is

IIT-JEE 2010 Paper 1 Offline

Let $$ABC$$ be a triangle such that $$\angle ACB = {\pi \over 6}$$ 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 $$c = 2x + 1$$ is (are)

IIT-JEE 2010 Paper 1 Offline

Let $$ABCD$$ be a quadrilateral with area $$18$$, with side $$AB$$ parallel to the side $$CD$$ and $$2AB=CD$$. Let $$AD$$ be perpendicular to $$AB$$ and $$CD$$. If a circle is drawn inside the quadrilateral $$ABCD$$ touching all the sides, then its radius is

IIT-JEE 2007

One angle of an isosceles $$\Delta $$ is $${120^ \circ }$$ and radius of its incircle $$ = \sqrt 3 $$. Then the area of the triangle in sq. units is

IIT-JEE 2006

In a triangle $$ABC$$, $$a,b,c$$ are the lengths of its sides and $$A,B,C$$ are the angles of triangle $$ABC$$. The correct relation is given by

IIT-JEE 2005 Screening

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

IIT-JEE 2005

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

IIT-JEE 2004 Screening

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

IIT-JEE 2003 Screening

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

IIT-JEE 2002 Screening

A man from the top of a $$100$$ metres high tower sees a car moving towards the tower at an angle of depression of $${30^ \circ }$$. After some time,the angle of depression becomes $${60^ \circ }$$. The distance (in metres) travelled by the car during this time is

IIT-JEE 2001 Screening

In a triangle $$ABC$$, $$2ac\,\sin {1 \over 2}\left( {A - B + C} \right) = $$

IIT-JEE 2000 Screening

In a triangle $$ABC$$, let $$\angle C = {\pi \over 2}$$. If $$r$$ is the inradius and $$R$$ is the circumradius of the triangle, then $$2(r+R)$$ is equal to

IIT-JEE 2000 Screening

A pole stands vertically inside a triangular park $$\Delta ABC$$. If the angle of elevation of the top of the pole from each corner of the park is same, then in $$\Delta ABC$$ the foot of the pole is at the

IIT-JEE 2000 Screening

If in a triangle $$PQR$$, $$\sin P,\sin Q,\sin R$$ are in $$A.P.,$$ then

IIT-JEE 1998

Let $${A_0}{A_1}{A_2}{A_3}{A_4}{A_5}$$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $${A_0}{A_1},{A_0}{A_2}$$ and $${A_0}{A_4}$$ is

IIT-JEE 1998

In a triangle $$ABC$$, $$\angle B = {\pi \over 3}$$ and $$\angle C = {\pi \over 4}$$. Let $$D$$ divide $$BC$$ internally in the ratio $$1:3$$ then $${{\sin \angle BAD} \over {\sin \angle CAD}}$$ is equal to

IIT-JEE 1995 Screening

If the lengths of the sides of triangle are $$3, 5, 7$$ then the largest angle of the triangle is

IIT-JEE 1994

In a triangle $$ABC$$, angle $$A$$ is greater than angle $$B$$. If the measures of angles $$A$$ and $$B$$ satify the equation $$3{\mathop{\rm sinx}\nolimits} - 4si{n^3}x - k = 0,$$ $$0 < k < 1$$, then the measure of angle $$C$$ is

IIT-JEE 1990

From the top of a light-house 60 metres high with its base at the sea-level, the angle of depression of a boat is $${15^ \circ }$$. The distance of the boat from the foot of the light house is

IIT-JEE 1983

If the bisector of the angle $$P$$ of a triangle $$PQR$$ meets $$QR$$ in $$S$$, then

IIT-JEE 1979

Subjective

If $${I_n}$$ is the area of $$n$$ sided regular polygon inscribed in a circle of unit radius and $${O_n}$$ be the area of the polygon circumscribing the given circle, prove that $$${I_n} = {{{O_n}} \over 2}\left( {1 + \sqrt {1 - {{\left( {{{2{I_n}} \over n}} \right)}^2}} } \right)$$$

IIT-JEE 2003

If $$\Delta $$ is the area of a triangle with side lengths $$a, b, c, $$ then show that $$\Delta \le {1 \over 4}\sqrt {\left( {a + b + c} \right)abc} $$. Also show that the equality occurs in the above inequality if and only if $$a=b=c$$.

IIT-JEE 2001

Let $$ABC$$ be a triangle with incentre $$I$$ and inradius $$r$$. Let $$D,E,F$$ be the feet of the perpendiculars from $$I$$ to the sides $$BC$$, $$CA$$ and $$AB$$ respectively. If $${r_1},{r_2}$$ and $${r_3}$$ are the radii of circles inscribed in the quadrilaterals $$AFIE$$, $$BDIF$$ and $$CEID$$ respectively, prove that $$${{{r_1}} \over {r - {r_1}}} + {{{r_2}} \over {r - {r_2}}} + {{{r_3}} \over {r - {r_3}}} = {{{r_1}{r_2}{r_3}} \over {\left( {e - {r_1}} \right)\left( {r - {r_2}} \right)\left( {r - {r_3}} \right)}}$$$

IIT-JEE 2000

Let $$ABC$$ be a triangle having $$O$$ and $$I$$ as its circumcenter and in centre respectively. If $$R$$ and $$r$$ are the circumradius and the inradius, respectively, then prove that $${\left( {IO} \right)^2} = {R^2} - 2{\mathop{\rm Rr}\nolimits} $$. Further show that the triangle BIO is a right-angled triangle if and only if $$b$$ is arithmetic mean of $$a$$ and $$c$$.

IIT-JEE 1999

A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose $${60^ \circ }$$ and $${30^ \circ }$$ are the maximum and the minimum angles of elevation of the bird and that they occur when the bird is at the points $$P$$ and $$Q$$ respectively on its path. Let $$\theta $$ be the angle of elevation of the bird when it is a point on the are of the circle exactly midway between $$P$$ and $$Q$$. Find the numerical value of $${\tan ^2}\theta $$. (Assume that the observer is not inside the vertical projection of the path of the bird.)

IIT-JEE 1998

Prove that a triangle $$ABC$$ is equilateral if and only if $$\tan A + \tan B + \tan C = 3\sqrt 3 $$.

IIT-JEE 1998

Let $${A_1},{A_2},........,{A_n}$$ be the vertices of an $$n$$-sided regular polygon such that $${1 \over {{A_1}{A_2}}} = {1 \over {{A_1}{A_3}}} + {1 \over {{A_1}{A_4}}}$$, Find the value of $$n$$.

IIT-JEE 1994

Consider the following statements connecting a triangle $$ABC$$

(i) The sides $$a, b, c$$ and area $$\Delta $$ are rational.

(ii) $$a,\tan {B \over 2},\tan {c \over 2}$$ are rational.

(iii) $$a,\sin A,\sin B,\sin C$$ are rational.
Prove that $$\left( i \right) \Rightarrow \left( {ii} \right) \Rightarrow \left( {iii} \right) \Rightarrow \left( i \right)$$

IIT-JEE 1994

A tower $$AB$$ leans towards west making an angle $$\alpha $$ with the vertical. The angular elevation of $$B$$, the topmost point of the tower is $$\beta $$ as observed from a point $$C$$ due west of $$A$$ at a distance $$d$$ from $$A$$. If the angular elevation of $$B$$ from a point $$D$$ due east of $$C$$ at a distance $$2d$$ from $$C$$ is $$\gamma $$, then prove that $$2$$ tan $$\alpha = - \cot \beta + \cot \gamma $$.

IIT-JEE 1994

An observer at $$O$$ notices that the angle of elevation of the top of a tower is $${30^ \circ }$$. The line joining $$O$$ to the base of the tower makes an angle of $${\tan ^{ - 1}}\left( {1/\sqrt 2 } \right)$$ with the North and is inclined Eastwards. The observer travels a distance of $$300$$ meters towards the North to a point A and finds the tower to his East. The angle of elevation of the top of the tower at $$A$$ is $$\phi $$, Find $$\phi $$ and the height of the tower.

IIT-JEE 1993

Three circles touch the one another externally. The tangent at their point of contact meet at a point whose distance from a point of contact is $$4$$. Find the ratio of the product of the radii to the sum of the radii of the circles.

IIT-JEE 1992

A man notices two objects in a straight line due west. After walking a distance $$c$$ due north he observes that the objects subtend an angle $$\alpha $$ at his eye; and, after walking a further distance $$2c$$ due north, an angle $$\beta $$. Show that the distance between the objects is $${{8c} \over {3\cot \beta - \cot \alpha }}$$; the height of the man is being ignored.

IIT-JEE 1991

In a triangle of base a the ratio of the other two sides is $$r\left( { < 1} \right)$$. Show that the altitude of the triangle is less than of equal to $${{ar} \over {1 - {r^2}}}$$

IIT-JEE 1991

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of the triangle.

IIT-JEE 1991

A vertical tower $$PQ$$ stands at a point $$P$$. Points $$A$$ and $$B$$ are located to the South and East of $$P$$ respectively. $$M$$ is the mid point of $$AB$$. $$PAM$$ is an equilateral triangle; and $$N$$ is the foot of the perpendicular from $$P$$ and $$AB$$. Let $$AN$$$$=20$$ mrtres and the angle of elevation of the top of the tower at $$N$$ is $${\tan ^{ - 1}}\left( 2 \right)$$. Determine the height of the tower and the angles of elevation of the top of the tower at $$A$$ and $$B$$.

IIT-JEE 1990

$$ABC$$ is a triangular park with $$AB=AC=100$$ $$m$$. A television tower stands at the midpoint of $$BC$$. The angles of elevetion of the top of the tower at $$A, B, C$$ are 45$$^ \circ $$, 60$$^ \circ $$, 60$$^ \circ $$, respectively. Find the height of the tower.

IIT-JEE 1989

A sign -post in the form of an isosceles triangle $$ABC$$ is mounted on a pole of height $$h$$ fixed to the ground. The base $$BC$$ of the triangle is parallel to the ground. A man standing on the ground at a distance $$d$$ from the sign-post finds that the top vertex $$A$$ of the triangle subtends an angle $$\beta $$ and either of the other two vertices subtends the same angle $$\alpha $$ at his feet. Find the area of the triangle.

IIT-JEE 1988

If in a triangle $$ABC$$, $$\cos A\cos B + \sin A\sin B\sin C = 1,$$ Show that $$a:b:c = 1:1:\sqrt 2 $$

IIT-JEE 1986

A ladder rests against a wall at an angle $$\alpha $$ to the horizintal. Its foot is pulled away from the wall through a distance $$a$$, so that it slides $$a$$ distance $$b$$ down the wall making an angle $$\beta $$ with the horizontal. Show that $$a = b\tan {1 \over 2}\left( {\alpha + \beta } \right)$$

IIT-JEE 1985

In a triangle $$ABC$$, the median to the side $$BC$$ is of length $$${1 \over {\sqrt {11 - 6\sqrt 3 } }}$$$ and it divides the angle $$A$$ into angles $${30^ \circ }$$ and $${45^ \circ }$$. Find the length of the side $$BC$$.

IIT-JEE 1985

For a triangle $$ABC$$ it is given that $$\cos A + \cos B + \cos C = {3 \over 2}$$. Prove that the triangle is equilateral.

IIT-JEE 1984

With usual notation, if in a triangle $$ABC$$;
$${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$$ then prove that $${{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}$$.

IIT-JEE 1984

The ex-radii $${r_1},{r_2},{r_3}$$ of $$\Delta $$$$ABC$$ are H.P. Show that its sides $$a, b, c$$ are in A.P.

IIT-JEE 1983

A vertical pole stands at a point $$Q$$ on a horizontal ground. $$A$$ and $$B$$ are points on the ground, $$d$$ meters apart. The pole subtends angles $$\alpha $$ and $$\beta $$ at $$A$$ and $$B$$ respectively. $$AB$$ subtends an angle $$\gamma $$ and $$Q$$. Find the height of the pole.

IIT-JEE 1982

Let the angles $$A, B, C$$ of a triangle $$ABC$$ be in A.P. and let $$b:c = \sqrt 3 :\sqrt 2 $$. Find the angle $$A$$.

IIT-JEE 1981

(i) $$PQ$$ is a vertical tower. $$P$$ is the foot and $$Q$$ is the top of the tower. $$A, B, C$$ are three points in the horizontal plane through $$P$$. The angles of elevation of $$Q$$ from $$A$$, $$B$$, $$C$$ are equal, and each is equal to $$\theta $$. The sides of the triangle $$ABC$$ are $$a, b, c$$; and the area of the triangle $$ABC$$ is $$\Delta $$. Show that the height of the tower is $${{abc\tan \theta } \over {4\Delta }}$$.

(ii) $$AB$$ is vertical pole. The end $$A$$ is on the level ground. $$C$$ is the middle point of $$AB$$. $$P$$ is a point on the level ground. The portion $$CB$$ subtends an angle $$\beta $$ at $$P$$. If $$AP = n\,AB,$$ then show that tan$$\beta $$ $$ = {n \over {2{n^2} + 1}}$$

IIT-JEE 1980

$$ABC$$ is a triangle with $$AB=AC$$. $$D$$ is any point on the side $$BC$$. $$E$$ and $$F$$ are points on the side $$AB$$ and $$AC$$, respectively, such that $$DE$$ is parallel to $$AC$$, and $$DF$$ is parallel to $$AB$$. Prove that $$$DF + FA + AE + ED = AB + AC$$$

IIT-JEE 1980

$$ABC$$ is a triangle. $$D$$ is the middle point of $$BC$$. If $$AD$$ is perpendicular to $$AC$$, then prove that $$$\cos A\,\cos C = {{2\left( {{c^2} - {a^2}} \right)} \over {3ac}}$$$

IIT-JEE 1980

(a) A balloon is observed simultaneously from three points $$A, B$$ and $$C$$ on a straight road directly beneath it. The angular elevation at $$B$$ is twice that at $$A$$ and the angular elevation at $$C$$ is thrice that at $$A$$. If the distance between $$A$$ and $$B$$ is a and the distance between $$B$$ and $$C$$ is $$b$$, find the height of the balloon in terms of $$a$$ and $$b$$.

(b) Find the area of the smaller part of a disc of radius $$10$$ cm, cut off by a chord $$AB$$ which subtends an angle of at the circumference.

IIT-JEE 1979

(a) If a circle is inscribed in a right angled triangle $$ABC$$ with the right angle at $$B$$, show that the diameter of the circle is equal to $$AB+BC-AC$$.

(b) If a triangle is inscribed in a circle, then the product of any two sides of the triangle is equal to the product of the diameter and the perpendicular distance of the third side from the opposite vertex. Prove the above statement.

IIT-JEE 1979

A triangle $$ABC$$ has sides $$AB=AC=5$$ cm and $$BC=6$$ cm Triangle $$A'B'C'$$ is the reflection of the triangle $$ABC$$ in a line parallel to $$AB$$ placed at a distance $$2$$ cm from $$AB$$, outside the triangle $$ABC$$. Triangle $$A''B''C''$$ is the reflection of the triangle $$A'B'C'$$ in a line parallel to $$BC$$ placed at a distance of $$2$$ cm from $$B'C'$$ outside the triangle $$A'B'C'$$. Find the distance between $$A$$ and $$A''$$.

IIT-JEE 1978