Properties of Triangle — JEE Main PYQs

If $\frac{\tan (\mathrm{A}-\mathrm{B})}{\tan \mathrm{A}}+\frac{{\sin }^2\mathrm{C}}{{\sin }^2\mathrm{A}}=1,\mathrm{A},\mathrm{B},\mathrm{C}\in \left(0,\frac{\pi }{2}\right)$, then

Let $\mathrm{A},\mathrm{B}$ be points on the two half-lines $x-\sqrt{3}|y|=\alpha ,\alpha >0$ at a distance of $\alpha$ from their point of intersection P . The line segment AB meets the angle bisector of the given half-lines at the point Q . If $\mathrm{PQ}=\frac{9}{2}$ and R is …

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

Let $\left(5,\frac{a}{4}\right)$ be the circumcenter of a triangle with vertices $A(a,-2)$, $B(a,6)$ 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 …

In the figure, ${\theta }_1+{\theta }_2=\frac{\pi }{2}$ and $\sqrt{3}(\mathrm{BE})=4(\mathrm{AB})$. If the area of $△\mathrm{CAB}$ is $2\sqrt{3}-3$ unit ${}^2$, when $\frac{{\theta }_2}{{\theta }_1}$ is the largest, then the perimeter (in unit) of …

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

If the line $\mathrm{x}=\mathrm{y}=\mathrm{z}$ intersects the line $\mathrm{x}\sin \mathrm{A}+\mathrm{y}\sin \mathrm{B}+\mathrm{z}\sin \mathrm{C}-18=0=\mathrm{x}\sin 2\mathrm{A}+\mathrm{y}\sin 2\mathrm{B}+\mathrm{z}\sin 2\mathrm{C}-9$, where …

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

Let $\mathrm{A}(\alpha ,-2),\mathrm{B}(\alpha ,6)$ and $\mathrm{C}\left(\frac{\alpha }{4},-2\right)$ be vertices of a $△\mathrm{ABC}$. If $\left(5,\frac{\alpha }{4}\right)$ is the circumcentre of $\Delta \mathrm{ABC}$, then which of the following is NOT correct about …

Let $\mathrm{a},\mathrm{b}$ and c be the length of sides of a triangle ABC such that $\frac{\mathrm{a}+\mathrm{b}}{7}=\frac{\mathrm{b}+\mathrm{c}}{8}=\frac{\mathrm{c}+\mathrm{a}}{9}$. If r and R are the radius of incircle and radius of circumcircle of the triangle ABC , resp …

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

Let in a right angled triangle, the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then $\sin \theta$ is equal to :

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

In $△ABC$, the lengths of sides $AC$ and $AB$ are 12 cm and 5 cm, respectively. If the area of $△ABC$ is $30{\mathrm{cm}}^2$ and $R$ and $r$ are respectively the radii of circumcircle and incircle of $△ABC$, then the value of $2R+r$ …

Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x+y=3$. If $R$ and $r$ be the radius of circumcircle and incircle respectively of $△ABC$, then $(R+r)$ is equal to :

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:

If $5,5r,5r^2$ are the lengths of the sides of a triangle, then $r$ cannot be equal to:

Consider a triangular plot ABC with sides $\mathrm{AB}=7\mathrm{m},\mathrm{BC}=5\mathrm{m}$ and $\mathrm{CA}=6\mathrm{m}$. A vertical lamp-post at the mid point D of AC subtends an angle $30^{\circ }$ at B. The height (in m) of the lamp-post is:

If the line $3x+4y-24=0$ intersects the $x$-axis at the point $A$ and the $y$-axis at the point $B$, then the incentre of the triangle $OAB$, where $O$ is the origin, is:

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

The angles $A,B$ and $C$ of a triangle $ABC$ are in $A.P$ and $a:b=1:\sqrt{3}$. If $c=4\mathrm{cm}$, then the area (in sq. cm ) 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 $x^2-c^2=y$, where $c$ is the length of the third side of the triangle, then the circumradius of the triangle is:

ABCD is a trapezium such that AB and CD are parallel and $\mathrm{BC}\perp \mathrm{CD}$. If $\angle \mathrm{ADB}=\theta ,\mathrm{BC}=\mathrm{p}$ and $\mathrm{CD}=\mathrm{q}$, then AB is equal to

In a $△PQR$, if $3\sin P+4\cos Q=6$ and $4\sin Q+3\cos P=1$, then the angle $R$ is equal to

In a triangle $\mathrm{PQR}$, $\angle \mathrm{R}=\frac{\pi }{2}$. If $\tan \left(\frac{\mathrm{P}}{2}\right)$ and $\tan \left(\frac{\mathrm{Q}}{2}\right)$ are the roots of $a{x}^2+bx+c=0,a\ne 0$ then

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

If in a triangle $ABC$, the altitudes from the vertices $A,B,C$ on opposite sides are in H.P., then $\sin A,\sin B,\sin C$ are in

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