JEE Main Properties of Triangle Practice Questions With Solutions

Two vertices of a triangle $\mathrm{ABC}$ABC are $\mathrm{A}(3,-1)$A \left(\right. 3 , - 1 \left.\right) and $\mathrm{B}(-2,3)$B \left(\right. - 2 , 3 \left.\right), and its orthocentre is $\mathrm{P}(1,1)$P \left(\right. 1 , 1 \left.\right). If the coordinates of the point $\mathrm{C}$C are $(\alpha ,\beta )$\left(\right. \alpha , \beta \left.\right) and the centre of the of the circle circumscribing the triangle $\mathrm{PAB}$PAB is $(\mathrm{h},\mathrm{k})$\left(\right. h , k \left.\right), then the value of $(\alpha +\beta )+2(\mathrm{h}+\mathrm{k})$\left(\right. \alpha + \beta \left.\right) + 2 \left(\right. \textrm{ } h + k \left.\right) equals

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

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

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

A straight line cuts off the intercepts $\mathrm{OA}=\mathrm{a}$OA = a and $\mathrm{OB}=\mathrm{b}$OB = b on the positive directions of $x$x-axis and $y$y axis respectively. If the perpendicular from origin $O$O to this line makes an angle of $\frac{\pi }{6}$\frac{\pi}{6} with positive direction of $y$y-axis and the area of $\mathrm{△}\mathrm{OAB}$\triangle OAB is $\frac{98}{3}\sqrt{3}$\frac{98}{3} \sqrt{3}, then ${\mathrm{a}}^2-{\mathrm{b}}^2$a^{2} - b^{2} is equal to :

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 $\frac{a+b}{7}=\frac{b+c}{8}=\frac{c+a}{9}$\frac{a + b}{7} = \frac{b + c}{8} = \frac{c + a}{9}. If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of $\frac{R}{r}$\frac{R}{r} is equal to :