There exists a triangle ABC satisfying

Important Questions on Properties of Triangles

HARD

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

In a non-right-angled triangle $\mathrm{\Delta }PQR,$ let $p\mathit{,}q\mathit{,}r$ 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 $p=\sqrt{3},q=1,$ and the radius of the circumcircle of the $\mathrm{\Delta }PQR$ equals $1,$ then which of the following options is/are correct?

HARD

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

Let $ABC$ be an acute angled triangle and let D be the midpoint of BC. If $AB=AD,$ then $\frac{\mathrm{tan }\left(B\right)}{\tan \left(C\right)}$ equals

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

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:

HARD

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

Let $ABC$ be an acute-angled triangle and let $D$ be the mid-point of $BC$. If $AB=AD$, then $\frac{\tan B}{\tan C}$ equal

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

In a triangle $ABC$ with $\angle A=90°$ , $P$ is a point on $BC$ such that $PA:PB=3:4.$ If $AB=\sqrt{7} \mathrm{and} AC=\sqrt{5}$ then $BP:PC$ is

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

If $a, b, c$ are in $\mathrm{GP}$ and $\log a-\log 2b, \log 2b-\log 3c$ and $\log 3c-\log a$ are in $\mathrm{AP}$, then $a, b$ and $c$ are the lengths of the sides of a triangle, which is

HARD

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

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 inradius to the circumradius of the triangle is

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

In a triangle $ABC$, if $A=2B$ and the sides opposite to the angles $A, B, C$ are $\alpha +1, \alpha -1$ and $\alpha$ respectively, then $\alpha =$

HARD

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

In a $∆PQR$, $P$ 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) of the side(s) of the triangle is (are)

HARD

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

The lengths of two adjacent sides of a cyclic quadrilateral are $2$ units and $5$ units and the angle between them is ${60}^{\mathrm{o}}$. If the area of the quadrilateral is $4\sqrt{3}$ sq. units, then the perimeter of the quadrilateral is

EASY

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

In any triangle $ABC$, $\left(b+c\right)\cos A+\left(c+a\right)\cos B+\left(a+b\right)\cos C$ equals

HARD

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

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

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

Let $a,b$ and $c$ be the lengths of the sides of a triangle with its opposite angles $A,B$ and $C$ respectively. If $a=3, b=4$ and $A={\sin }^{-1}\left(\frac{3}{4}\right),$ then the angle $B$ is

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

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

HARD

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

In a $∆ABC$, if $b=10$, $a{\cos }^2\frac{C}{2}+c{\cos }^2\frac{A}{2}=15$ and the area of the triangle is $15\sqrt{3}$ sq. units, then $\cot \frac{B}{2}=$

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

In $△ABC$, if $a=5$ and $\tan \frac{A-B}{2}=\frac{1}{4}\tan \frac{A+B}{2}$, then $\sqrt{a^2-b^2}=$

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

In $\mathrm{\Delta }ABC$, right-angled at $A$, the circumradius, inradius and radius of the excircle opposite to $A$ are respectively in the ratio $2:5:\lambda$, then the roots of the equation $x^2-\left(\lambda -5\right)x+\left(\lambda -6\right)=0$ are

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

Let $ABC$ be a triangle such that $AB=BC.$ Let $F$ be the midpoint of $AB$ and $X$ be a point on $BC$ such that $FX$ is perpendicular to $AB.$ If $BX=3XC$ then the ratio $\frac{BC}{AC}$ equals,

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

In $\mathrm{\Delta }ABC$, if ${\sin }^{2}A+{\sin }^{2}B={\sin }^{2}C$ and $l\left(AB\right)=10$, then the maximum value of the area of $\mathrm{\Delta }ABC$ is

MEDIUM

Mathematics>Trigonometry>Properties of Triangles>Relations Between Sides and Angles of a Triangle

With the usual notation in $∆ABC$, if $\angle A+\angle B=120°, a=\sqrt{3}+1$ units and $b=\sqrt{3}-1$ units, then the ratio $\angle A:\angle B$ is