Mathematics
Properties of Triangle
Previous Years Questions

Let $P Q R S$ be a quadrilateral in a plane, where $Q R=1, \angle P Q R=\angle Q R S=70^{\circ}, \angle P Q S=15^{\circ}$ and $\angle P R S=40^{\circ}... 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 ... 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, respectivel... 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 ... 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 ... In a triangle $$PQR$$, $$P$$ is the largest angle and $$\cos P = {1 \over 3}$$. Further the incircle of the triangle touches the sides $$PQ$$, $$QR$$ ... 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 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 ... There exists a triangle $$ABC$$ satisfying the conditions ## Numerical 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 _________. In a triangle PQR, let a = QR, b = RP, and c = PQ. If |a| = 3, |b| = 4 and $${{a\,.(\,c - \,b)} \over {c\,.\,(a - \,b)}} = {{|a|} \over {|a| + |b|}}$$... 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 $$... Let ABC and ABC' be two non-congruent triangles with sides AB = 4, AC = AC' = 2$$\sqrt2$$and angle B = 30$$^\circ$$. The absolute value of the differ... ## MCQ (Single Correct Answer) In a triangle the sum of two sides is$$x$$and the product of the same sides is$$y$$. If$${x^2} - {c^2} = y$$, where$$c$$is the third side of the... Let$$PQR$$be a triangle of area$$\Delta $$with$$a=2$$,$$b = {7 \over 2}$$and$$c = {5 \over 2}$$; where$$a, b,$$and$$c$$are the lengths of ... If the angles$$A, B$$and$$C$$of a triangle are in an arithmetic progression and if$$a, b$$and$$c$$denote the lengths of the sides opposite to ... Let$$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$...
Let $$ABCD$$ be a quadrilateral with area $$18$$, with side $$AB$$ parallel to the side $$CD$$ and $$2AB=CD$$. Let $$AD$$ be perpendicular to $$AB$$ a...
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
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 t...
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
The sides of a triangle are in the ratio $$1:\sqrt 3 :2$$, then the angles of the triangle are in the ratio
If the angles of a triangle are in the ratio $$4:1:1$$, then the ratio of the longest side to the perimeter is
Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $$ABC$$ ($$R$$ being the radius of the circumcircle)?
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,t...
In a triangle $$ABC$$, $$2ac\,\sin {1 \over 2}\left( {A - B + C} \right) =$$
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 e...
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 sam...
If in a triangle $$PQR$$, $$\sin P,\sin Q,\sin R$$ are in $$A.P.,$$ then
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...
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 $... If the lengths of the sides of triangle are $$3, 5, 7$$ then the largest angle of the triangle is In a triangle $$ABC$$, angle $$A$$ is greater than angle $$B$$. If the measures of angles $$A$$ and $$B$$ satify the equation $$3{\mathop{\rm sinx}\no... From the top of a light-house 60 metres high with its base at the sea-level, the angle of depression of a boat is$${15^ \circ }$$. The distance of th... If the bisector of the angle$$P$$of a triangle$$PQR$$meets$$QR$$in$$S$$, then ## Subjective If$${I_n}$$is the area of$$n$$sided regular polygon inscribed in a circle of unit radius and$${O_n}$$be the area of the polygon circumscribing t... If$$\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}... 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 $$BC$$, $$CA... Let$$ABC$$be a triangle having$$O$$and$$I$$as its circumcenter and in centre respectively. If$$R$$and$$r$$are the circumradius and the inrad... A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose$${60^ \circ }$$and$${30^ \circ }$$are the max... Prove that a triangle$$ABC$$is equilateral if and only if$$\tan A + \tan B + \tan C = 3\sqrt 3 $$. Let$${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 ... Consider the following statements connecting a triangle $$ABC$$ (i) The sides $$a, b, c$$ and area $$\Delta$$ are rational. (ii) $$a,\tan {B \over 2... A tower$$AB$$leans towards west making an angle$$\alpha $$with the vertical. The angular elevation of$$B$$, the topmost point of the tower is$$\... 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 ma... 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 $$4$$.... 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 $$\alpha... In a triangle of base a the ratio of the other two sides is$$r\left( { < 1} \right)$$. Show that the altitude of the triangle is less than of equa... The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of the triangle. A vertical tower$$PQ$$stands at a point$$P$$. Points$$A$$and$$B$$are located to the South and East of$$P$$respectively.$$M$$is the mid poin...$$ABC$$is a triangular park with$$AB=AC=100m$$. A television tower stands at the midpoint of$$BC$$. The angles of elevetion of the top of the ... A sign -post in the form of an isosceles triangle$$ABC$$is mounted on a pole of height$$h$$fixed to the ground. The base$$BC$$of the triangle is... If in a triangle$$ABC$$,$$\cos A\cos B + \sin A\sin B\sin C = 1,$$Show that$$a:b:c = 1:1:\sqrt 2 $$A ladder rests against a wall at an angle$$\alpha $$to the horizintal. Its foot is pulled away from the wall through a distance$$a$$, so that it sl... In a triangle$$ABC$$, the median to the side$$BC$$is of length$$${1 \over {\sqrt {11 - 6\sqrt 3 } }}$$and it divides the angle$$A$$into angle... For a triangle$$ABC$$it is given that$$\cos A + \cos B + \cos C = {3 \over 2}$$. Prove that the triangle is equilateral. With usual notation, if in a triangle$$ABC$$;$${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$$then prove that$${{\cos A} \o...
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.
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...
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 $$A$$.
$$ABC$$ is a triangle. $$D$$ is the middle point of $$BC$$. If $$AD$$ is perpendicular to $$AC$$, then prove that $$\cos A\,\cos C = {{2\left( {{c^2...$$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$$, respectively, s... (i)$$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$... (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 $$AB+B... (a) A balloon is observed simultaneously from three points$$A, B$$and$$C$$on a straight road directly beneath it. The angular elevation at$$B$$i... A triangle$$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$$... ## Fill in the Blanks In a triangle$$ABC$$,$$a:b:c=4:5:6$$. The ratio of the radius of the circumcircle to that of the incircle is ............ In a triangle$$ABC$$,$$AD$$is the altitude from$$A$$. Given$$b>c$$,$$\angle C = {23^ \circ }$$and$$AD = {{abc} \over {{b^2} - {c^2}}}$$the... A circle is inscribed in an equilateral triangle of side$$a$$. The area of any square inscribed in this circle is .............. If in a triangle$$ABC$$,$${{2\cos A} \over a} + {{\cos B} \over b} + {{2\cos C} \over c} = {a \over {bc}} + {b \over {ca}},$$then the value of the ... If the angles of a triangle are$${30^ \circ }$$and$${45^ \circ }$$and the included side is$$\left( {\sqrt 3 + 1} \right)$$cms, then the area of... A polygon of nine sides, each of length$$2$$, is inscribed in a circle. The radius of the circle is ................. The set of all real numbers$$a$$such that$${a^2} + 2a,2a + 3$$and$${a^2} + 3a + 8$$are the sides of a triangle is ........... In a triangle$$ABC$$, if cot$$A$$, cot$$B$$, cot$$C$$are in A.P., then$${a^2},{b^2},{c^2}$$, are in ............... progression.$$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$$and ...$$ABC$$is a triangle,$$P$$is a point on$$AB$$, and$$Q$$is point on$$AC$$such that$$\angle AQP = \angle ABC$$. Complete the relation$$${{are...
In a $$\Delta ABC,\,\angle A = {90^ \circ }$$ and $$AD$$ is an altitude. Complete the relation {{BD} \over {BA}} = {{AB} \over {\left( {....} \right...
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Joint Entrance Examination