Properties of Triangle
116 Questions
2005
JEE Advanced
MCQ
IIT-JEE 2005 Screening
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
A.
$\left( {b - c} \right)\sin \left( {{{B - C} \over 2}} \right) = a\cos {A \over 2}$
B.
$\left( {b - c} \right)cos\left( {{A \over 2}} \right) = a\,sin{{B - C} \over 2}$
C.
$\left( {b + c} \right)\sin \left( {{{B + C} \over 2}} \right) = a\cos {A \over 2}$
D.
$\left( {b - c} \right)cos\left( {{A \over 2}} \right) = 2a\,sin{{B + C} \over 2}$
2005
JEE Advanced
MCQ
IIT-JEE 2005
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 + 2\sqrt 3 $
B.
$6 + 4\sqrt 3 $
C.
$12 + {{7\sqrt 3 } \over 4}$
D.
$3 + {{7\sqrt 3 } \over 4}$
2004
JEE Mains
MCQ
AIEEE 2004
The sides of a triangle are $\sin \alpha ,\,\cos \alpha $ and $\sqrt {1 + \sin \alpha \cos \alpha } $ for some $0 < \alpha < {\pi \over 2}$. Then the greatest angle of the triangle is :
A.
${150^ \circ }$
B.
${90^ \circ }$
C.
${120^ \circ }$
D.
${60^ \circ }$
2004
JEE Advanced
MCQ
IIT-JEE 2004 Screening
The sides of a triangle are in the ratio $1:\sqrt 3 :2$, then the angles of the triangle are in the ratio
A.
$1:3:5$
B.
$2:3:4$
C.
$3:2:1$
D.
$1:2:3$
2003
JEE Mains
MCQ
AIEEE 2003
If in a $\Delta ABC$ $a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2},$ then the sides $a, b$ and $c$ :
A.
satisfy $a+b=c$
B.
are in A.P
C.
are in G.P
D.
are in H.P
2003
JEE Mains
MCQ
AIEEE 2003
In a triangle $ABC$, medians $AD$ and $BE$ are drawn. If $AD=4$,
$\angle DAB = {\pi \over 6}$ and $\angle ABE = {\pi \over 3}$, then the area of the $\angle \Delta ABC$ is :
$\angle DAB = {\pi \over 6}$ and $\angle ABE = {\pi \over 3}$, then the area of the $\angle \Delta ABC$ is :
A.
${{64} \over 3}$
B.
${8 \over 3}$
C.
${{16} \over 3}$
D.
${{32} \over {3\sqrt 3 }}$
2003
JEE Mains
MCQ
AIEEE 2003
The sum of the radii of inscribed and circumscribed circles for an $n$ sided regular polygon of side $a, $ is :
A.
${a \over 4}\cot \left( {{\pi \over {2n}}} \right)$
B.
$a\cot \left( {{\pi \over {n}}} \right)$
C.
${a \over 2}\cot \left( {{\pi \over {2n}}} \right)$
D.
$a\cot \left( {{\pi \over {2n}}} \right)$
2003
JEE Advanced
MCQ
IIT-JEE 2003 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
A.
$\sqrt 3 :\left( {2 + \sqrt 3 } \right)$
B.
$1:6$
C.
$1:2 + \sqrt 3 $
D.
$2:3$
2003
JEE Advanced
Numerical
IIT-JEE 2003
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
$${I_n} = {{{O_n}} \over 2}\left( {1 + \sqrt {1 - {{\left( {{{2{I_n}} \over n}} \right)}^2}} } \right)$$
Correct Answer: Solve it.
2002
JEE Mains
MCQ
AIEEE 2002
The sides of a triangle are $3x + 4y,$ $4x + 3y$ and $5x + 5y$ where $x$, $y>0$ then the triangle is :
A.
right angled
B.
obtuse angled
C.
equilateral
D.
none of these
2002
JEE Mains
MCQ
AIEEE 2002
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 $b > c$
2002
JEE Advanced
MCQ
IIT-JEE 2002 Screening
Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $ABC$ ($R$ being the radius of the circumcircle)?
A.
$a,\,\sin \,A,sin\,B$
B.
$a,b,c$
C.
$a,\,\sin \,B,R$
D.
$a,\,\sin \,A,R$
2001
JEE Advanced
MCQ
IIT-JEE 2001 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
A.
$100\sqrt 3 $
B.
$200\sqrt 3 /3$
C.
$100\sqrt 3 /3$
D.
$200\sqrt 3 $
2001
JEE Advanced
Numerical
IIT-JEE 2001
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 $a=b=c$.
Correct Answer: Solve it.
2000
JEE Advanced
MCQ
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
A.
$a+b$
B.
$b+c$
C.
$c+a$
D.
$a+b+c$
2000
JEE Advanced
MCQ
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
A.
centroid
B.
circumcentre
C.
incentre
D.
orthocentre
2000
JEE Advanced
MCQ
IIT-JEE 2000 Screening
In a triangle $ABC$, $2ac\,\sin {1 \over 2}\left( {A - B + C} \right) = $
A.
${a^2} + {b^2} - {c^2}$
B.
${c^2} + {a^2} - {b^2}$
C.
${b^2} - {c^2} - {a^2}$
D.
${c^2} - {a^2} - {b^2}$
2000
JEE Advanced
Numerical
IIT-JEE 2000
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
$${{{r_1}} \over {r - {r_1}}} + {{{r_2}} \over {r - {r_2}}} + {{{r_3}} \over {r - {r_3}}} = {{{r_1}{r_2}{r_3}} \over {\left( {e - {r_1}} \right)\left( {r - {r_2}} \right)\left( {r - {r_3}} \right)}}$$
Correct Answer: Solve it.
1999
JEE Advanced
Numerical
IIT-JEE 1999
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 $c$.
Correct Answer: Solve it.
1998
JEE Advanced
MCQ
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
A.
${3 \over 4}$
B.
$3\sqrt 3 $
C.
$3$
D.
${{3\sqrt 3 } \over 2}$
1998
JEE Advanced
MCQ
IIT-JEE 1998
If in a triangle $PQR$, $\sin P,\sin Q,\sin R$ are in $A.P.,$ then
A.
the altitudes are in $A.P.$
B.
the altitudes are in $H.P.$
C.
the medians are in $G.P.$
D.
the medians are in $A.P$
1998
JEE Advanced
Numerical
IIT-JEE 1998
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.)
Correct Answer: $${3 \over 5}$$
1998
JEE Advanced
Numerical
IIT-JEE 1998
Prove that a triangle $ABC$ is equilateral if and only if $\tan A + \tan B + \tan C = 3\sqrt 3 $.
Correct Answer: Solve it.
1996
JEE Advanced
Numerical
IIT-JEE 1996
In a triangle $ABC$, $a:b:c=4:5:6$. The ratio of the radius of the circumcircle to that of the incircle is ............
Correct Answer: $$16:7$$
1995
JEE Advanced
MCQ
IIT-JEE 1995 Screening
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
A.
${1 \over {\sqrt 6 }}$
B.
${1 \over 3}$
C.
${1 \over {\sqrt 3 }}$
D.
$\sqrt {{2 \over 3}} $
1994
JEE Advanced
MCQ
IIT-JEE 1994
If the lengths of the sides of triangle are $3, 5, 7$ then the largest angle of the triangle is
A.
${\pi \over 2}$
B.
${5\pi \over 6}$
C.
${2\pi \over 3}$
D.
${3\pi \over 4}$
1994
JEE Advanced
Numerical
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 $.
Correct Answer: Solve it.
1994
JEE Advanced
Numerical
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 $\left( i \right) \Rightarrow \left( {ii} \right) \Rightarrow \left( {iii} \right) \Rightarrow \left( i \right)$
Correct Answer: Solve it.
1994
JEE Advanced
Numerical
IIT-JEE 1994
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 $n$.
Correct Answer: $$7$$
1994
JEE Advanced
Numerical
IIT-JEE 1994
A circle is inscribed in an equilateral triangle of side $a$. The area of any square inscribed in this circle is ..............
Correct Answer: $${{{a^2}} \over 6}$$ sq. unit
1994
JEE Advanced
Numerical
IIT-JEE 1994
In a triangle $ABC$, $AD$ is the altitude from $A$. Given $b>c$, $\angle C = {23^ \circ }$ and $AD = {{abc} \over {{b^2} - {c^2}}}$ then $\angle B = $.................
Correct Answer: $${113^ \circ }$$
1993
JEE Advanced
Numerical
IIT-JEE 1993
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.
Correct Answer: $${45^ \circ }$$, $${150\sqrt 2 }$$ $$m$$
1993
JEE Advanced
Numerical
IIT-JEE 1993
If in a triangle $ABC$, ${{2\cos A} \over a} + {{\cos B} \over b} + {{2\cos C} \over c} = {a \over {bc}} + {b \over {ca}},$ then the value of the angle $A$ is .................... degrees.
Correct Answer: $${90^ \circ }$$
1992
JEE Advanced
Numerical
IIT-JEE 1992
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.
Correct Answer: $$16:1$$
1991
JEE Advanced
Numerical
IIT-JEE 1991
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.
Correct Answer: Solve it.
1991
JEE Advanced
Numerical
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 ${{ar} \over {1 - {r^2}}}$
Correct Answer: Solve it.
1991
JEE Advanced
Numerical
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.
Correct Answer: $$4, 5, 6$$
1990
JEE Advanced
MCQ
IIT-JEE 1990
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
A.
${\pi \over 3}$
B.
${\pi \over 2}$
C.
${2\pi \over 3}$
D.
${5\pi \over 6}$
1990
JEE Advanced
Numerical
IIT-JEE 1990
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$$=20$ mrtres and the angle of elevation of the top of the tower at $N$ is ${\tan ^{ - 1}}\left( 2 \right)$. Determine the height of the tower and the angles of elevation of the top of the tower at $A$ and $B$.
Correct Answer: $$40\sqrt 3 \,\,\,m,\,\,\,{60^ \circ },\,\,\,{45^ \circ }$$
1989
JEE Advanced
Numerical
IIT-JEE 1989
$ABC$ is a triangular park with $AB=AC=100$ $m$. A television tower stands at the midpoint of $BC$. The angles of elevetion of the top of the tower at $A, B, C$ are 45$^ \circ $, 60$^ \circ $, 60$^ \circ $, respectively. Find the height of the tower.
Correct Answer: $$50\sqrt 3 $$ $$m$$
1988
JEE Advanced
Numerical
IIT-JEE 1988
A sign -post in the form of an isosceles triangle $ABC$ is mounted on a pole of height $h$ fixed to the ground. The base $BC$ of the triangle is parallel to the ground. A man standing on the ground at a distance $d$ from the sign-post finds that the top vertex $A$ of the triangle subtends an angle $\beta $ and either of the other two vertices subtends the same angle $\alpha $ at his feet. Find the area of the triangle.
Correct Answer: $$\left( {d\tan \beta - h} \right)\sqrt {{h^2}{{\cot }^2}\alpha - {d^2}} $$
1988
JEE Advanced
Numerical
IIT-JEE 1988
If the angles of a triangle are ${30^ \circ }$ and ${45^ \circ }$ and the included side is $\left( {\sqrt 3 + 1} \right)$ cms, then the area of the triangle is ...............
Correct Answer: $${{\sqrt 3 + 1} \over 2}$$ sq. unit
1987
JEE Advanced
MSQ
IIT-JEE 1987
In a triangle, the lengths of the two larger sides are $10$ and $9$, respectively. If the angles are in $AP$. Then the length of the third side can be
A.
$5 - \sqrt 6 $
B.
$3\sqrt 3 $
C.
$5$
D.
$5 + \sqrt 6 $
1987
JEE Advanced
Numerical
IIT-JEE 1987
A polygon of nine sides, each of length $2$, is inscribed in a circle. The radius of the circle is .................
Correct Answer: $$\cos ec\,{\raise0.5ex\hbox{$\scriptstyle \pi $}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 9$}}$$
1986
JEE Advanced
MSQ
IIT-JEE 1986
There exists a triangle $ABC$ satisfying the conditions
A.
$b\sin A = a,A < \pi /2$
B.
$b\sin A > a,A > \pi /2$
C.
$b\sin A > a,A < \pi /2$
D.
$b\sin A < a,A < \pi /2,b > a$
1986
JEE Advanced
Numerical
IIT-JEE 1986
If in a triangle $ABC$, $\cos A\cos B + \sin A\sin B\sin C = 1,$ Show that $a:b:c = 1:1:\sqrt 2 $
Correct Answer: Solve it.
1985
JEE Advanced
Numerical
IIT-JEE 1985
In a triangle $ABC$, the median to the side $BC$ is of length
$${1 \over {\sqrt {11 - 6\sqrt 3 } }}$$ and it divides the angle $A$ into angles ${30^ \circ }$ and ${45^ \circ }$. Find the length of the side $BC$.
Correct Answer: $$2$$ units.
1985
JEE Advanced
Numerical
IIT-JEE 1985
A ladder rests against a wall at an angle $\alpha $ to the horizintal. Its foot is pulled away from the wall through a distance $a$, so that it slides $a$ distance $b$ down the wall making an angle $\beta $ with the horizontal. Show that $a = b\tan {1 \over 2}\left( {\alpha + \beta } \right)$
Correct Answer: Solve it.
1985
JEE Advanced
Numerical
IIT-JEE 1985
In a triangle $ABC$, if cot $A$, cot $B$, cot $C$ are in A.P., then ${a^2},{b^2},{c^2}$, are in ............... progression.
Correct Answer: Arithmetic
1985
JEE Advanced
Numerical
IIT-JEE 1985
The set of all real numbers $a$ such that ${a^2} + 2a,2a + 3$ and ${a^2} + 3a + 8$ are the sides of a triangle is ...........
Correct Answer: $$\left( {5,x} \right)$$