Properties of Triangle
116 Questions
1984
JEE Advanced
Numerical
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 ${{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}$.
${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$ then prove that ${{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}$.
Correct Answer: Solve it.
1984
JEE Advanced
Numerical
IIT-JEE 1984
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
JEE Advanced
MCQ
IIT-JEE 1983
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
JEE Advanced
Numerical
IIT-JEE 1983
The ex-radii ${r_1},{r_2},{r_3}$ of $\Delta $$ABC$ are H.P. Show that its sides $a, b, c$ are in A.P.
Correct Answer: Solve it.
1982
JEE Advanced
Numerical
IIT-JEE 1982
A vertical pole stands at a point $Q$ on a horizontal ground. $A$ and $B$ are points on the ground, $d$ meters apart. The pole subtends angles $\alpha $ and $\beta $ at $A$ and $B$ respectively. $AB$ subtends an angle $\gamma $ and $Q$. Find the height of the pole.
Correct Answer: $${d \over {\sqrt {{{\cot }^2}\alpha + {{\cot }^2}\beta - \cot \alpha \cot \beta \cot \gamma } }}$$
1981
JEE Advanced
Numerical
IIT-JEE 1981
Let the angles $A, B, C$ of a triangle $ABC$ be in A.P. and let $b:c = \sqrt 3 :\sqrt 2 $. Find the angle $A$.
Correct Answer: $${75^ \circ }$$
1980
JEE Advanced
Numerical
IIT-JEE 1980
$ABC$ is a triangle with $AB=AC$. $D$ is any point on the side $BC$. $E$ and $F$ are points on the side $AB$ and $AC$, respectively, such that $DE$ is parallel to $AC$, and $DF$ is parallel to $AB$. Prove that
$$DF + FA + AE + ED = AB + AC$$
Correct Answer: Solve it.
1980
JEE Advanced
Numerical
IIT-JEE 1980
(i) $PQ$ is a vertical tower. $P$ is the foot and $Q$ is the top of the tower. $A, B, C$ are three points in the horizontal plane through $P$. The angles of elevation of $Q$ from $A$, $B$, $C$ are equal, and each is equal to $\theta $. The sides of the triangle $ABC$ are $a, b, c$; and the area of the triangle $ABC$ is $\Delta $. Show that the height of the tower is ${{abc\tan \theta } \over {4\Delta }}$.
(ii) $AB$ is vertical pole. The end $A$ is on the level ground. $C$ is the middle point of $AB$. $P$ is a point on the level ground. The portion $CB$ subtends an angle $\beta $ at $P$. If $AP = n\,AB,$ then show that tan$\beta $ $ = {n \over {2{n^2} + 1}}$
Correct Answer: (i) Solve it.
<p>(ii) Solve it.</p>
1980
JEE Advanced
Numerical
IIT-JEE 1980
$ABC$ is a triangle. $D$ is the middle point of $BC$. If $AD$ is perpendicular to $AC$, then prove that
$$\cos A\,\cos C = {{2\left( {{c^2} - {a^2}} \right)} \over {3ac}}$$
Correct Answer: Solve it.
1980
JEE Advanced
Numerical
IIT-JEE 1980
$ABC$ is a triangle, $P$ is a point on $AB$, and $Q$ is point on $AC$ such that $\angle AQP = \angle ABC$. Complete the relation
$${{area\,\,of\,\,\Delta APQ} \over {area\,\,of\,\,\Delta ABC}} = {{\left( {...} \right)} \over {A{C^2}}}$$
Correct Answer: $$A{P^2}$$
1980
JEE Advanced
Numerical
IIT-JEE 1980
$ABC$ is a triangle with $\angle B$ greater than $\angle C.\,D$ and $E$ are points on $BC$ such that $AD$ is perpendicular to $BC$ and $AE$ is the bisector of angle $A$. Complete the relation
$$\angle DAE = {1 \over 2}\left[ {\left( {} \right) - \angle C} \right]$$
Correct Answer: $${\angle B}$$
1980
JEE Advanced
Numerical
IIT-JEE 1980
In a $\Delta ABC,\,\angle A = {90^ \circ }$ and $AD$ is an altitude. Complete the relation ${{BD} \over {BA}} = {{AB} \over {\left( {....} \right)}}$.
Correct Answer: $$B$$ $$C$$
1979
JEE Advanced
MCQ
IIT-JEE 1979
If the bisector of the angle $P$ of a triangle $PQR$ meets $QR$ in $S$, then
A.
$QS=SR$
B.
$QS:SR$ $= PR:PQ$
C.
$QS:SR=PQ:PR$
D.
None of these
1979
JEE Advanced
Numerical
IIT-JEE 1979
(a) A balloon is observed simultaneously from three points $A, B$ and $C$ on a straight road directly beneath it. The angular elevation at $B$ is twice that at $A$ and the angular elevation at $C$ is thrice that at $A$. If the distance between $A$ and $B$ is a and the distance between $B$ and $C$ is $b$, find the height of the balloon in terms of $a$ and $b$.
(b) Find the area of the smaller part of a disc of radius $10$ cm, cut off by a chord $AB$ which subtends an angle of at the circumference.
Correct Answer: (a) $${a \over {2b}}\sqrt {\left( {a + b} \right)\left( {3b - a} \right)} $$
<p>(b) $$3.91$$ sq. cm.</p>
1979
JEE Advanced
Numerical
IIT-JEE 1979
(a) If a circle is inscribed in a right angled triangle $ABC$ with the right angle at $B$, show that the diameter of the circle is equal to $AB+BC-AC$.
(b) If a triangle is inscribed in a circle, then the product of any two sides of the triangle is equal to the product of the diameter and the perpendicular distance of the third side from the opposite vertex. Prove the above statement.
Correct Answer: Solve it.
1978
JEE Advanced
Numerical
IIT-JEE 1978
A triangle $ABC$ has sides $AB=AC=5$ cm and $BC=6$ cm Triangle $A'B'C'$ is the reflection of the triangle $ABC$ in a line parallel to $AB$ placed at a distance $2$ cm from $AB$, outside the triangle $ABC$. Triangle $A''B''C''$ is the reflection of the triangle $A'B'C'$ in a line parallel to $BC$ placed at a distance of $2$ cm from $B'C'$ outside the triangle $A'B'C'$. Find the distance between $A$ and $A''$.
Correct Answer: $$8\sqrt {{{17} \over 5}} $$