Properties of Triangle — JEE Advanced PYQs

Consider an obtuse angled triangle ABC in which the difference between the largest and the smallest angle is $\frac{\pi }{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.

Let a be the area of the triangle …

Consider an obtuse angled triangle ABC in which the difference between the largest and the smallest angle is $\frac{\pi }{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.

Then the inradius of the triangle …

Let PQRS be a quadrilateral in a plane, where $QR=1$, $\angle PQR=\angle QRS=70^{\circ }$, $\angle PQS=15^{\circ }$ and $\angle PRS=40^{\circ }$. If $\angle RPS={\theta }^{\circ }$, $PQ=\alpha$ and $PS=\beta$, then the interval(s) that contain(s) the value of …

In a triangle ABC, let $AB=\sqrt{23},BC=3$ and $CA=4$. Then the value of $\frac{\cot A+\cot C}{\cot B}$ is ____.

Consider a triangle PQR having sides of lengths p, q and r opposite to the angles P, Q and R, respectively. Then which of the following statements is (are) TRUE?

Let x, y and z be positive real numbers. Suppose x, y and z are lengths of the sides of a triangle opposite to its angles X, Y and Z, respectively. If

then which of the following statements is/are TRUE?

In a non-right-angled triangle $\Delta \mathrm{PQR}$, let p, q, r denote the lengths of the sides opposite to the angles at P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E , and RS and PE intersect at O . If …

In a triangle $PQR$, let $\angle PQR=30^{\circ }$ and the sides $PQ$ and $QR$ have lengths $10\sqrt{3}$ and 10, respectively. Then, which of the following statement(s) is (are) TRUE?

In a triangle $XYZ$, let $x,y,z$ be the lengths of sides opposite to the angles $X,Y,Z$, respectively, and $2s=x+y+z$. If $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$ and area of incircle of the triangle $XYZ$ is $\frac{8\pi }{3}$, then

Match the entries in Column I with the entries in Column II.

Column I

(A) In a triangle $△XYZ$, let $a,b$ and $c$ be the lengths of the sides opposite to the angles $X,Y$ and $Z$, respectively. If $2\left(a^2-b^2\right)=c^2$ and …

In a triangle the sum of two sides is $x$ and the product of the same two sides is $y$. If $x^2-c^2=y$, where $c$ is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is

In a triangle $PQR,P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the incircle of the triangle touches the sides $PQ,QR$ and $RP$ at $N,L$ and $M$ respectively, such that the lengths of $PN,QL$ and $RM$ are consecutive even integers. Then possible length(s) …

Let $PQR$ be a triangle of area $△$ with $a=2,b=\frac{7}{2}$ and $c=\frac{5}{2}$, where $a,b$ and $c$ are the lengths of the sides of the triangle opposite to the angles at $P,Q$ and $R$ respectively. Then $\frac{2\sin P-\sin 2P}{2\sin P+\sin 2P}$ equals

Consider a triangle ABC and let a, b and c denote the lengths of the sides opposite to vertices A, B and C respectively. Suppose $\mathrm{a}=6,\mathrm{b}=10$ and the area of the triangle is $15\sqrt{3}$. If $\angle \mathrm{ACB}$ is obtuse and if r denotes the radius of t …

If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of the sides opposite to A, B and C respectively, then the value of the expression $\frac{a}{c}\sin 2C+\frac{c}{a}\sin 2A$ is

Let $ABC$ and $AB{C}^{\prime }$ be two non-congruent triangles with sides $AB=4$, $AC=A{C}^{\prime }=2\sqrt{2}$ and angle $B=30^{\circ }$. The absolute value of the difference between the areas of these triangles is

In a triangle $ABC$ with fixed base $BC$, the vertex $A$ moves such that

If $a,b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A,B$ and $C$, respectively, then

Consider the circle $x^2+y^2=9$ and the parabola $y^2=8x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $x$-axis at $R$ and tangents to the parabola at $P$ and $Q$ …

Given an isosceles triangle, whose one angle is 120° and radius of its incircle $=\sqrt{3}$. Then the area of the triangle in sq. units is

Internal bisector of $\angle A$ of triangle ABC meets side BC at D. A line drawn through D perpendicular to AD intersects the side AC at E and the side AB at F. If a, b, c represent sides of $△ABC$ then