Properties of Triangle

In a triangle \(ABC\), angle \(A\) is greater than angle \(B\). If the measures of angles \(A\) and \(B\) satify the equation \(3{\mathop{\rm sinx}\nolimits} - 4si{n^3}x - k = 0,\) \(0 < k < 1\), then the measure of angle \(C\) is

\(ABC\) 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. Fin...

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-po...

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 ...............

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 polygon of nine sides, each of length \(2\), is inscribed in a circle. The radius of the circle is .................

There exists a triangle \(ABC\) satisfying the conditions

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 slides \(a\) distance \(b\) down the wall making an angle \(\beta\) with the horizontal. Sh...

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 \(ABC\), 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 \(BC\).

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.

With usual notation, if in a triangle \(ABC\);
\({{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}\) then prove that \({{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}\).

For a triangle \(ABC\) it is given that \(\cos A + \cos B + \cos C = {3 \over 2}\). Prove that the triangle is equilateral.

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

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 \(\alpha\) and \(\beta\) at \(A\) and \(B\) respectively. \(AB\) subtends an angle $$\g...

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 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...

\(ABC\) 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 {...

\(ABC\) is a triangle. \(D\) is the middle point of \(BC\). If \(AD\) is perpendicular to \(AC\), then prove that
\($\cos A\,\cos C = {{2\left( {{c^2} - {a^2}} \right)} \over {3ac}}\)$

\(ABC\) 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}\l...

In a \(\Delta ABC,\,\angle A = {90^ \circ }\) and \(AD\) is an altitude. Complete the relation \({{BD} \over {BA}} = {{AB} \over {\left( {....} \right)}}\).

(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 ...

If the bisector of the angle \(P\) of a triangle \(PQR\) meets \(QR\) in \(S\), then

(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 ...

(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 d...

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'...