Properties of Triangle

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 :

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 :

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 :

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

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 :

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 $\frac{\pi}{6}$ with positive direction of $y$-axis and the area of $\triangle \mathrm{OAB}$ is $\frac{98}{3} \sqrt{3}$, then $\mathrm{a}^{2}-\mathrm{b}^{2}$ is equal to :

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 $\gamma$ denote the perimeter of the triangle. Then $\alpha+\beta+\gamma$ is

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 circle circumscribing the triangle $\mathrm{PAB}$ is $(\mathrm{h}, \mathrm{k})$, then the value of $(\alpha+\beta)+2(\mathrm{~h}+\mathrm{k})$ equals

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

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 $c$ :

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 :

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 :

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 :

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 $a{x^2} + bx + 1 = 0$, then the value of ${\alpha ^2} + {\beta ^2} - \alpha \beta$ 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 :

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