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Properties of Triangle

116 Questions Start JEE Mains Test
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET BITSAT
1984 Q101 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 ${{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}$.
1984 Q102 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.
1983 Q103 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 Q104 JEE Advanced Numerical
14 Mar 2026
The ex-radii ${r_1},{r_2},{r_3}$ of $\Delta $$ABC$ are H.P. Show that its sides $a, b, c$ are in A.P.
1982 Q105 JEE Advanced Numerical
14 Mar 2026
A vertical pole stands at a point $Q$ on a horizontal ground. $A$ and $B$ are points on the ground, $d$ meters apart. The pole subtends angles $\alpha $ and $\beta $ at $A$ and $B$ respectively. $AB$ subtends an angle $\gamma $ and $Q$. Find the height of the pole.
1981 Q106 JEE Advanced Numerical
14 Mar 2026
Let the angles $A, B, C$ of a triangle $ABC$ be in A.P. and let $b:c = \sqrt 3 :\sqrt 2 $. Find the angle $A$.
1980 Q107 JEE Advanced Numerical
14 Mar 2026
$ABC$ is a triangle with $AB=AC$. $D$ is any point on the side $BC$. $E$ and $F$ are points on the side $AB$ and $AC$, respectively, such that $DE$ is parallel to $AC$, and $DF$ is parallel to $AB$. Prove that $$DF + FA + AE + ED = AB + AC$$
1980 Q108 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}}$

1980 Q109 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 $$\cos A\,\cos C = {{2\left( {{c^2} - {a^2}} \right)} \over {3ac}}$$
1980 Q110 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 $${{area\,\,of\,\,\Delta APQ} \over {area\,\,of\,\,\Delta ABC}} = {{\left( {...} \right)} \over {A{C^2}}}$$
1980 Q111 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 $$\angle DAE = {1 \over 2}\left[ {\left( {} \right) - \angle C} \right]$$
1980 Q112 JEE Advanced Numerical
14 Mar 2026
In a $\Delta ABC,\,\angle A = {90^ \circ }$ and $AD$ is an altitude. Complete the relation ${{BD} \over {BA}} = {{AB} \over {\left( {....} \right)}}$.
1979 Q113 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 Q114 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.

1979 Q115 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.

1978 Q116 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 $A''$.