Properties of Triangle
78 Questions
Start JEE Advanced Test
1990
Q51
JEE Advanced
Numerical
14 Mar 2026
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 }$$
1989
Q52
JEE Advanced
Numerical
14 Mar 2026
$ABC$ is a triangular park with $AB=AC=100$ $m$. A television tower stands at the midpoint of $BC$. The angles of elevetion of the top of the tower at $A, B, C$ are 45$^ \circ $, 60$^ \circ $, 60$^ \circ $, respectively. Find the height of the tower.
Correct Answer: $$50\sqrt 3 $$ $$m$$
1988
Q53
JEE Advanced
Numerical
14 Mar 2026
A sign -post in the form of an isosceles triangle $ABC$ is mounted on a pole of height $h$ fixed to the ground. The base $BC$ of the triangle is parallel to the ground. A man standing on the ground at a distance $d$ from the sign-post finds that the top vertex $A$ of the triangle subtends an angle $\beta $ and either of the other two vertices subtends the same angle $\alpha $ at his feet. Find the area of the triangle.
Correct Answer: $$\left( {d\tan \beta - h} \right)\sqrt {{h^2}{{\cot }^2}\alpha - {d^2}} $$
1988
Q54
JEE Advanced
Numerical
14 Mar 2026
If the angles of a triangle are ${30^ \circ }$ and ${45^ \circ }$ and the included side is $\left( {\sqrt 3 + 1} \right)$ cms, then the area of the triangle is ...............
Correct Answer: $${{\sqrt 3 + 1} \over 2}$$ sq. unit
1987
Q55
JEE Advanced
MSQ
14 Mar 2026
In a triangle, the lengths of the two larger sides are $10$ and $9$, respectively. If the angles are in $AP$. Then the length of the third side can be
A.
$5 - \sqrt 6 $
B.
$3\sqrt 3 $
C.
$5$
D.
$5 + \sqrt 6 $
1987
Q56
JEE Advanced
Numerical
14 Mar 2026
A polygon of nine sides, each of length $2$, is inscribed in a circle. The radius of the circle is .................
Correct Answer: $$\cos ec\,{\raise0.5ex\hbox{$\scriptstyle \pi $}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 9$}}$$
1986
Q57
JEE Advanced
MSQ
14 Mar 2026
There exists a triangle $ABC$ satisfying the conditions
A.
$b\sin A = a,A < \pi /2$
B.
$b\sin A > a,A > \pi /2$
C.
$b\sin A > a,A < \pi /2$
D.
$b\sin A < a,A < \pi /2,b > a$
1986
Q58
JEE Advanced
Numerical
14 Mar 2026
If in a triangle $ABC$, $\cos A\cos B + \sin A\sin B\sin C = 1,$ Show that $a:b:c = 1:1:\sqrt 2 $
Correct Answer: Solve it.
1985
Q59
JEE Advanced
Numerical
14 Mar 2026
In a triangle $ABC$, the median to the side $BC$ is of length
$${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$$ units.
1985
Q60
JEE Advanced
Numerical
14 Mar 2026
A ladder rests against a wall at an angle $\alpha $ to the horizintal. Its foot is pulled away from the wall through a distance $a$, so that it slides $a$ distance $b$ down the wall making an angle $\beta $ with the horizontal. Show that $a = b\tan {1 \over 2}\left( {\alpha + \beta } \right)$
Correct Answer: Solve it.
1985
Q61
JEE Advanced
Numerical
14 Mar 2026
In a triangle $ABC$, if cot $A$, cot $B$, cot $C$ are in A.P., then ${a^2},{b^2},{c^2}$, are in ............... progression.
Correct Answer: Arithmetic
1985
Q62
JEE Advanced
Numerical
14 Mar 2026
The set of all real numbers $a$ such that ${a^2} + 2a,2a + 3$ and ${a^2} + 3a + 8$ are the sides of a triangle is ...........
Correct Answer: $$\left( {5,x} \right)$$
1984
Q63
JEE Advanced
Numerical
14 Mar 2026
With usual notation, if in a triangle $ABC$;
${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$ then prove that ${{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}$.
${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$ then prove that ${{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}$.
Correct Answer: Solve it.
1984
Q64
JEE Advanced
Numerical
14 Mar 2026
For a triangle $ABC$ it is given that $\cos A + \cos B + \cos C = {3 \over 2}$. Prove that the triangle is equilateral.
Correct Answer: Solve it.
1983
Q65
JEE Advanced
MCQ
14 Mar 2026
From the top of a light-house 60 metres high with its base at the sea-level, the angle of depression of a boat is ${15^ \circ }$. The distance of the boat from the foot of the light house is
A.
$\left( {{{\sqrt 3 - 1} \over {\sqrt 3 + 1}}} \right)60$ metres
B.
$\left( {{{\sqrt 3 + 1} \over {\sqrt 3 - 1}}} \right)60$ metres
C.
${\left( {{{\sqrt 3 + 1} \over {\sqrt 3 - 1}}} \right)^2}$ metres
D.
none of these
1983
Q66
JEE Advanced
Numerical
14 Mar 2026
The ex-radii ${r_1},{r_2},{r_3}$ of $\Delta $$ABC$ are H.P. Show that its sides $a, b, c$ are in A.P.
Correct Answer: Solve it.
1982
Q67
JEE Advanced
Numerical
14 Mar 2026
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.
Correct Answer: $${d \over {\sqrt {{{\cot }^2}\alpha + {{\cot }^2}\beta - \cot \alpha \cot \beta \cot \gamma } }}$$
1981
Q68
JEE Advanced
Numerical
14 Mar 2026
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 $A$.
Correct Answer: $${75^ \circ }$$
1980
Q69
JEE Advanced
Numerical
14 Mar 2026
$ABC$ 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
$$DF + FA + AE + ED = AB + AC$$
Correct Answer: Solve it.
1980
Q70
JEE Advanced
Numerical
14 Mar 2026
(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}}$
Correct Answer: (i) Solve it.
<p>(ii) Solve it.</p>
1980
Q71
JEE Advanced
Numerical
14 Mar 2026
$ABC$ is a triangle. $D$ is the middle point of $BC$. If $AD$ is perpendicular to $AC$, then prove that
$$\cos A\,\cos C = {{2\left( {{c^2} - {a^2}} \right)} \over {3ac}}$$
Correct Answer: Solve it.
1980
Q72
JEE Advanced
Numerical
14 Mar 2026
$ABC$ is a triangle, $P$ is a point on $AB$, and $Q$ is point on $AC$ such that $\angle AQP = \angle ABC$. Complete the relation
$${{area\,\,of\,\,\Delta APQ} \over {area\,\,of\,\,\Delta ABC}} = {{\left( {...} \right)} \over {A{C^2}}}$$
Correct Answer: $$A{P^2}$$
1980
Q73
JEE Advanced
Numerical
14 Mar 2026
$ABC$ is a triangle with $\angle B$ greater than $\angle C.\,D$ and $E$ are points on $BC$ such that $AD$ is perpendicular to $BC$ and $AE$ is the bisector of angle $A$. Complete the relation
$$\angle DAE = {1 \over 2}\left[ {\left( {} \right) - \angle C} \right]$$
Correct Answer: $${\angle B}$$
1980
Q74
JEE Advanced
Numerical
14 Mar 2026
In a $\Delta ABC,\,\angle A = {90^ \circ }$ and $AD$ is an altitude. Complete the relation ${{BD} \over {BA}} = {{AB} \over {\left( {....} \right)}}$.
Correct Answer: $$B$$ $$C$$
1979
Q75
JEE Advanced
MCQ
14 Mar 2026
If the bisector of the angle $P$ of a triangle $PQR$ meets $QR$ in $S$, then
A.
$QS=SR$
B.
$QS:SR$ $= PR:PQ$
C.
$QS:SR=PQ:PR$
D.
None of these
1979
Q76
JEE Advanced
Numerical
14 Mar 2026
(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.
Correct Answer: (a) $${a \over {2b}}\sqrt {\left( {a + b} \right)\left( {3b - a} \right)} $$
<p>(b) $$3.91$$ sq. cm.</p>
1979
Q77
JEE Advanced
Numerical
14 Mar 2026
(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.
Correct Answer: Solve it.
1978
Q78
JEE Advanced
Numerical
14 Mar 2026
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 $A''$.
Correct Answer: $$8\sqrt {{{17} \over 5}} $$