Properties of Triangle

Let the area of a $\triangle P Q R$ with vertices $P(5,4), Q(-2,4)$ and $R(a, b)$ be 35 square units. If its orthocenter and centroid are $O\left(2, \frac{14}{5}\right)$ and $C(c, d)$ respectively, then $c+2 d$ is equal to

Let $\mathrm{A}(6,8), \mathrm{B}(10 \cos \alpha,-10 \sin \alpha)$ and $\mathrm{C}(-10 \sin \alpha, 10 \cos \alpha)$, be the vertices of a triangle. If $L(a, 9)$ and $G(h, k)$ be its orthocenter and centroid respectively, then $(5 a-3 h+6 k+...

Two vertices of a triangle \(\mathrm{ABC}\) are \(\mathrm{A}(3,-1)\) and \(\mathrm{B}(-2,3)\), and its orthocentre is \(\mathrm{P}(1,1)\). If the coordinates of the point \(\mathrm{C}\) are \((\alpha, \beta)\) and the centre of the of the c...

In a triangle \(\mathrm{ABC}, \mathrm{BC}=7, \mathrm{AC}=8, \mathrm{AB}=\alpha \in \mathrm{N}\) and \(\cos \mathrm{A}=\frac{2}{3}\). If \(49 \cos (3 \mathrm{C})+42=\frac{\mathrm{m}}{\mathrm{n}}\), where \(\operatorname{gcd}(m, n)=1\), then ...

Consider a triangle \(\mathrm{ABC}\) having the vertices \(\mathrm{A}(1,2), \mathrm{B}(\alpha, \beta)\) and \(\mathrm{C}(\gamma, \delta)\) and angles \(\angle A B C=\frac{\pi}{6}\) and \(\angle B A C=\frac{2 \pi}{3}\). If the points $$\math...

Let \(\left(5, \frac{a}{4}\right)\) be the circumcenter of a triangle with vertices \(\mathrm{A}(a,-2), \mathrm{B}(a, 6)\) and \(C\left(\frac{a}{4},-2\right)\). Let \(\alpha\) denote the circumradius, \(\beta\) denote the area and $$\gamma$...

A straight line cuts off the intercepts \(\mathrm{OA}=\mathrm{a}\) and \(\mathrm{OB}=\mathrm{b}\) on the positive directions of \(x\)-axis and \(y\) axis respectively. If the perpendicular from origin \(O\) to this line makes an angle of $$...

For a triangle \(ABC\), the value of \(\cos 2A + \cos 2B + \cos 2C\) is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?

If the line $x=y=z$ intersects the line

$x \sin A+y \sin B+z \sin C-18=0=x \sin 2 A+y \sin 2 B+z \sin 2 C-9$,

where $A, B, C$ are the angles of a triangle $A B C$, then $80\left(\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}\right)$

...

In a triangle ABC, if \(\cos \mathrm{A}+2 \cos \mathrm{B}+\cos C=2\) and the lengths of the sides opposite to the angles A and C are 3 and 7 respectively, then \(\mathrm{\cos A-\cos C}\) is equal to

In the figure, \(\theta_{1}+\theta_{2}=\frac{\pi}{2}\) and \(\sqrt{3}(\mathrm{BE})=4(\mathrm{AB})\). If the area of \(\triangle \mathrm{CAB}\) is \(2 \sqrt{3}-3\) unit \({ }^{2}\), when \(\frac{\theta_{2}}{\theta_{1}}\) is the largest, then...

The lengths of the sides of a triangle are 10 + x2, 10 + x2 and 20 \(-\) 2x2. If for x = k, the area of the triangle is maximum, then 3k2 is equal to :

Let a, b and c be the length of sides of a triangle ABC such that \({{a + b} \over 7} = {{b + c} \over 8} = {{c + a} \over 9}\). If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the va...

Let \({{\sin A} \over {\sin B}} = {{\sin (A - C)} \over {\sin (C - B)}}\), where A, B, C are angles of triangle ABC. If the lengths of the sides opposite these angles are a, b, c respectively, then :

The triangle of maximum area that can be inscribed in a given circle of radius 'r' is :

If a rectangle is inscribed in an equilateral triangle of side length \(2\sqrt 2\) as shown in the figure, then the square of the largest area of such a rectangle is _____________.

Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be \(\left( {{{10} \over 3},{7 \over 3}} \right)\). If \(\alpha\), \(\beta\) are the roots of the equation $$a{x^2} + bx + 1 ...

If in a triangle ABC, AB = 5 units, \(\angle B = {\cos ^{ - 1}}\left( {{3 \over 5}} \right)\) and radius of circumcircle of \(\Delta\)ABC is 5 units, then the area (in sq. units) of \(\Delta\)ABC is :

Let ABCD be a square of side of unit length. Let a circle C1 centered at A with unit radius is drawn. Another circle C2 which touches C1 and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the ci...

In \(\Delta\)ABC, the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of \(\Delta\)ABC is 30 cm2 and R and r are respectively the radii of circumcircle and incircle of \(\Delta\)ABC, then the value of 2R + r (in cm)...

A triangle ABC lying in the first quadrant has two vertices as A(1, 2) and B(3, 1). If \(\angle BAC = {90^o}\) and area\(\left( {\Delta ABC} \right) = 5\sqrt 5\) s units, then the abscissa of the vertex C is :

If the lengths of the sides of a triangle are in A.P.
and the greatest angle is double the smallest, then
a ratio of lengths of the sides of this triangle is :

A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (–1, 1) and (2, 3). Then the centroid of this triangle is :

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x2 – c2 = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :

Given \({{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}\) for a \(\Delta\)ABC with usual notation.
If   \({{\cos A} \over \alpha } = {{\cos B} \over \beta } = {{\cos C} \over \gamma },\) then the ordered triad ($$\alpha...

With the usual notation, in \(\Delta\)ABC, if \(\angle A + \angle B\) = 120o, a = \(\sqrt 3\) \(+\) 1, b = \(\sqrt 3\) \(-\) 1 then the ratio \(\angle A:\angle B,\) is :

The angles A, B and C of a triangle ABC are in A.P. and a : b = 1 : \(\sqrt 3\). If c = 4 cm, then the area (in sq. cm)
of this triangle is :

Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre
of this triangle, then the radius of the circle having line segment AC as diameter, is :

In a \(\Delta PQR,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu}\) If \(3{\mkern 1mu} \sin {\mkern 1mu} P + 4{\mkern 1mu} \cos {\mkern 1mu} Q = 6\) and \(4\sin Q + 3\cos P = 1,\) then the angle R is equal to :

For a regular polygon, let \(r\) and \(R\) be the radii of the inscribed and the circumscribed circles. A \(false\) statement among the following is :

If in a \(\Delta ABC\), the altitudes from the vertices \(A, B, C\) on opposite sides are in H.P, then \(\sin A,\sin B,\sin C\) are in :

In a triangle \(ABC\), let \(\angle C = {\pi \over 2}\). If \(r\) is the inradius and \(R\) is the circumradius of the triangle \(ABC\), then \(2(r+R)\) equals :

The sides of a triangle are \(\sin \alpha ,\,\cos \alpha\) and \(\sqrt {1 + \sin \alpha \cos \alpha }\) for some \(0 < \alpha < {\pi \over 2}\). Then the greatest angle of the triangle is :

The sum of the radii of inscribed and circumscribed circles for an \(n\) sided regular polygon of side \(a,\) is :

If in a \(\Delta ABC\) \(a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2},\) then the sides \(a, b\) and \(c\) :

In a triangle \(ABC\), medians \(AD\) and \(BE\) are drawn. If \(AD=4\),
\(\angle DAB = {\pi \over 6}\) and \(\angle ABE = {\pi \over 3}\), then the area of the \(\angle \Delta ABC\) is :

The sides of a triangle are \(3x + 4y,\) \(4x + 3y\) and \(5x + 5y\) where \(x\), \(y>0\) then the triangle is :

In a triangle with sides \(a, b, c,\) \({r_1} > {r_2} > {r_3}\) (which are the ex-radii) then :