Properties of Triangle · Mathematics · JEE Main

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 coor...

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...

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 respectiv...

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 ...

A straight line cuts off the intercepts $$\mathrm{OA}=\mathrm{a}$$ and $$\mathrm{OB}=\mathrm{b}$$ on the positive directions of $$x$$-axis and $$y$$ a...

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 ra...

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 oppos...

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, ...

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

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 \ove...

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) =...

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 :

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 :

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 tria...

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

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 o...

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

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...

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...

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 ...

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 :

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...

The sides of a triangle are $$\sin \alpha ,\,\cos \alpha $$ and $$\sqrt {1 + \sin \alpha \cos \alpha } $$ for some $$0 < \alpha < {\pi \over 2...

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

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 th...

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...

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 :

In a triangle $$\mathrm{ABC}, \mathrm{BC}=7, \mathrm{AC}=8, \mathrm{AB}=\alpha \in \mathrm{N}$$ and $$\cos \mathrm{A}=\frac{2}{3}$$. If $$49 \cos (3 \...

Consider a triangle $$\mathrm{ABC}$$ having the vertices $$\mathrm{A}(1,2), \mathrm{B}(\alpha, \beta)$$ and $$\mathrm{C}(\gamma, \delta)$$ and angles ...

If the line $x=y=z$ intersects the line $x \sin A+y \sin B+z \sin C-18=0=x \sin 2 A+y \sin 2 B+z \sin 2 C-9$, where $A, B, C$ are the angles of a tr...

In the figure, $$\theta_{1}+\theta_{2}=\frac{\pi}{2}$$ and $$\sqrt{3}(\mathrm{BE})=4(\mathrm{AB})$$. If the area of $$\triangle \mathrm{CAB}$$ is $$2 ...

If a rectangle is inscribed in an equilateral triangle of side length $$2\sqrt 2 $$ as shown in the figure, then the square of the largest area of suc...

In $$\Delta$$ABC, the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of $$\Delta$$ABC is 30 cm2 and R and r are respectively...

Let ABCD be a square of side of unit length. Let a circle C1 centered at A with unit radius is drawn. Another circle C2 which touches C1 and the lines...