Properties of Triangle

\(\text { Let } a \text { be the area of the triangle } A B C \text {. Then the value of }(64 a)^2 \text { is }\) :

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 (are) TRUE?

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 _________.

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, respectively. If\(\tan {X \over 2} + \tan {Z \over 2} = {{2y} \over {x + y + z}}\), then which of the...

In a triangle PQR, let a = QR, b = RP, and c = PQ. If |a| = 3, |b| = 4 and \({{a\,.(\,c - \,b)} \over {c\,.\,(a - \,b)}} = {{|a|} \over {|a| + |b|}}\), then the value of |a \(\times\) b|2 is ......

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 from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and...

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 \({{s - x} \over 4} = {{s - y} \over 3} = {{s - z} \over 2}\) and area of incircle of the ...

Match the following :

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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 triangle, then the ratio of the in radius to the circum-radius of the triangle is

In a triangle \(PQR\), \(P\) is the largest angle and \(\cos P = {1 \over 3}\). Further the incircle of the triangle touches the sides \(PQ\), \(QR\) and \(RP\) at \(N,L\) and \(M\) respectively, such that the lengths of \(PN, QL\) and $$RM...

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 the sides of the triangle opposite to the angles at \(P.Q\) and \(R\) respectively. Then $...

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 \(a = 6,b = 10\) and the area of the triangle is \(15\sqrt 3\), if \(\angle ACB\) is obtuse...

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 \(A\), \(B\) and \(C\) respectively. The value(s) of \(x\) for which $$a = {x^2} + x + 1,\,\,\,b = {x...

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 \(A, B\) and \(C\) respectively, then the value of the expression $${a \over c}\sin 2C + ...

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 difference between the areas of these triangles is ___________.

Let \(ABCD\) be a quadrilateral with area \(18\), with side \(AB\) parallel to the side \(CD\) and \(2AB=CD\). Let \(AD\) be perpendicular to \(AB\) and \(CD\). If a circle is drawn inside the quadrilateral \(ABCD\) touching all the sides, ...

Internal bisector of $\angle A$ of triangle $A B C$ meets side BC at D . A line drawn through D perpendicular to AD intersects the side AC at E and the side AB at F . If $a, b, c$ represent sides of $\triangle \mathrm{ABC}$ then

Given an isosceles triangle, whose one angle is $120^{\circ}$ and radius of its incircle $=\sqrt{3}$. Then the area of the triangle in sq. units is

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

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

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

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( {...

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,the angle of depression becomes \({60^ \circ }\). The distance (in metres) travelled by the...

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 \(a=b=c\).

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 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

In a triangle \(ABC\), \(2ac\,\sin {1 \over 2}\left( {A - B + C} \right) =\)

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 insc...

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}\n...

If in a triangle \(PQR\), \(\sin P,\sin Q,\sin R\) are in \(A.P.,\) then

Prove that a triangle \(ABC\) is equilateral if and only if \(\tan A + \tan B + \tan C = 3\sqrt 3\).

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 ...

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

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 \(ABC\), \(\angle B = {\pi \over 3}\) 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

If the lengths of the sides of triangle are \(3, 5, 7\) then the largest angle of the triangle is

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 \(A\). I...

A circle is inscribed in an equilateral triangle of side \(a\). The area of any square inscribed in this circle is ..............

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 th...

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 \(n\).

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 \(\angle B =\).................

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 inclin...

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.

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.

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 ...

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.

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 \({{ar} \over {1 - {r^2}}}\)

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 perpendicula...