iCON Education - Properties of Triangle JEE Mains Test

Let the area of a $\triangle P Q R$ with vertices $P(5,4), Q(-2,4)$ and $R(a, b)$ be 35 square units. If its orthocenter and centroid are $O\left(2, \frac{14}{5}\right)$ and $C(c, d)$ respectively, then $c+2 d$ is equal to

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

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

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

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

$\overrightarrow {MA} \,.\,\overrightarrow {MB} = - 18$

$\sin 2A + \sin 2B + \sin 2C = \sin A + \sin B + \sin C$

perimeter of $\Delta ABC$ is 18$\sqrt3$

area of $\Delta ABC$ is ${{27\sqrt 3 } \over 2}$

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 :

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

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 :

b² $-$ a² = a² + c²

b², c², a² are in A.P.

c², a², b² are in A.P.

a², b², c² are in A.P.

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 :

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

An equilateral triangle having each of its side of length $\sqrt 3 $r.

An equilateral triangle of height ${{2r} \over 3}$.

A right angle triangle having two of its sides of length 2r and r.

An isosceles triangle with base equal to 2r.

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 :

${{69} \over {256}}$

${{71} \over {256}}$

$ - {{71} \over {256}}$

$ - {{69} \over {256}}$

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 :

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 :

$\left( {{1 \over 3},2} \right)$

$\left( {{1 \over 3},{5 \over 3}} \right)$

$\left( {1,{7 \over 3}} \right)$

$\left( {{1 \over 3},1} \right)$

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 :

${4 \over {\sqrt 3 }}$

${2 \over {\sqrt 3 }}$

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 :

Given ${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$ for a $\Delta $ABC with usual notation.

If ${{\cos A} \over \alpha } = {{\cos B} \over \beta } = {{\cos C} \over \gamma },$ then the ordered triad ($\alpha $, $\beta $, $\gamma $) has a value :

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 :

${y \over {\sqrt 3 }}$

${c \over {\sqrt 3 }}$

With the usual notation, in $\Delta $ABC, if $\angle A + \angle B$ = 120^o, a = $\sqrt 3 $ $+$ 1, b = $\sqrt 3 $ $-$ 1 then the ratio $\angle A:\angle B,$ is :

Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is :

${{3\sqrt 5 } \over 2}$

$3\sqrt {{5 \over 2}} $

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 :

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 :

There is a regular polygon with ${r \over R} = {1 \over {\sqrt 2 }}$

There is a regular polygon with ${r \over R} = {2 \over 3}$

There is a regular polygon with ${r \over R} = {{\sqrt 3 } \over 2}$

There is a regular polygon with ${r \over R} = {1 \over 2}$

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

If in a $\Delta ABC$, 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 $0 < \alpha < {\pi \over 2}$. Then the greatest angle of the triangle 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 $c$ :

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 :

${{32} \over {3\sqrt 3 }}$

The sum of the radii of inscribed and circumscribed circles for an $n$ sided regular polygon of side $a, $ is :

${a \over 4}\cot \left( {{\pi \over {2n}}} \right)$

$a\cot \left( {{\pi \over {n}}} \right)$

${a \over 2}\cot \left( {{\pi \over {2n}}} \right)$

$a\cot \left( {{\pi \over {2n}}} \right)$

The sides of a triangle are $3x + 4y,$ $4x + 3y$ and $5x + 5y$ where $x$, $y>0$ then the triangle is :

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