Properties of Triangles · Mathematics · MHT CET

In a triangle ABC , with usual notations. $\frac{2 \cos \mathrm{~A}}{\mathrm{a}}+\frac{\cos \mathrm{B}}{\mathrm{b}}+\frac{2 \cos \mathrm{C}}{\mathrm{c}}=\frac{\mathrm{a}}{\mathrm{bc}}+\frac{\mathrm{b}}{\mathrm{ca}}$. Then $\angle \mathrm{A}=$

If in triangle ABC , with usual notations $\sin \frac{\mathrm{A}}{2} \cdot \sin \frac{\mathrm{C}}{2}=\sin \frac{\mathrm{B}}{2}$ and 2 s is the perimeter of the triangle, then the value of $s$ is

Let $\mathrm{A} \equiv(0,0), \mathrm{B}(3,0), \mathrm{C}(0,-4)$ are vertices of $\triangle A B C$, then the co-ordinates of incentre of $\triangle \mathrm{ABC}$ is

In a triangle with one of the angles $120^{\circ}$, the lengths of the sides form an A.P. If length of the greatest side is 7 m , then the area of the triangle is

In $\triangle \mathrm{ABC}$, with usual notations, if $\cos \frac{B}{2}=\sqrt{\frac{c+a}{2 a}}$, then $a^2=$

In a triangle ABC with usual notations, $\cot \frac{\mathrm{A}}{2}+\cot \frac{\mathrm{B}}{2}+\cot \frac{\mathrm{C}}{2}=$

With usual notations, in $\triangle \mathrm{ABC}$, the lengths of two sides are 10 cm and 9 cm respectively. If angles $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are in A.P. then perimeter of $\triangle \mathrm{ABC}$ is

In a triangle ABC , with usual notations, if $\mathrm{a}=5, \mathrm{~b}=4, \cos (\mathrm{~A}-\mathrm{B})=\frac{31}{32}$, then $\mathrm{c}=$

In a triangle ABC with usual notations if, $\cot \frac{A}{2}=\frac{b+c}{a}$, then the triangle $A B C$ is

In a triangle ABC with usual notations, if $3 \mathrm{a}=\mathrm{b}+\mathrm{c}$, then $\cot \frac{\mathrm{B}}{2} \cdot \cot \frac{\mathrm{C}}{2}=$

In $\triangle A B C$, with usual notations, if $\mathrm{a}^4+\mathrm{b}^4+\mathrm{c}^4-2 \mathrm{a}^2 \mathrm{c}^2-2 \mathrm{c}^2 \mathrm{~b}^2=0$, then $\angle \mathrm{C}=\ldots$

In a triangle ABC with usual notations if, $\tan \left(\frac{\mathrm{B}-\mathrm{C}}{2}\right)=x \cot \frac{\mathrm{~A}}{2}$, then $x=$

With usual notations, the perimeter of a triangle ABC is 6 times the arithmetic mean of sine of its angles. If $\mathrm{a}=1$, then $\angle \mathrm{A}=$

In a triangle ABC , with usual notations, $\tan \left(\frac{\mathrm{A}}{2}\right)=\frac{5}{6}, \tan \left(\frac{\mathrm{C}}{2}\right)=\frac{2}{5}$, then

With usual notation, in triangle ABC , $\mathrm{m} \angle \mathrm{A}=30^{\circ}$ then the value of $\left(1+\frac{\mathrm{a}}{\mathrm{c}}+\frac{\mathrm{b}}{\mathrm{c}}\right)\left(1+\frac{\mathrm{c}}{\mathrm{b}}-\frac{\mathrm{a}}{\mathrm{b}}\right)$ is equal to

In $\triangle A B C$, with usual notations, $a \cos B=b \cos A, a \cos C \neq c \cos A$ then $\mathrm{A}(\triangle \mathrm{ABC})$ $\qquad$ sq. units.

In a triangle ABC , with usual notations if $\mathrm{a}=4, \mathrm{~b}=8, \angle \mathrm{C}=60^{\circ}$, then the value of $\angle \mathrm{B}$ and the ratio $\cos \mathrm{A}: \cos \mathrm{C}$ respectively are,

In a triangle ABC with usual notations if $|\overline{\mathrm{BC}}|=8,|\overline{\mathrm{CA}}|=7,|\overline{\mathrm{AB}}|=10$ then the projection of $\overline{\mathrm{AB}}$ on $\overline{\mathrm{AC}}$ is

In a triangle ABC , the sides $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are such that they are the roots of the equation $x^3-11 x^2+38 x-40=0$ Then

$$ \frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}= $$

In a triangle ABC with usual notations if $\mathrm{a}=13$, $b=14, c=15$ Then $\sin A=$

In a triangle $A B C$, with usual notations, $3 \mathrm{~b}=\mathrm{a}+\mathrm{c}$, then $\cot \frac{\mathrm{A}}{2} \cdot \cot \frac{\mathrm{C}}{2}=$

In a triangle ABC , with usual notations if $\frac{2 \cos \mathrm{~A}}{\mathrm{a}}+\frac{\cos \mathrm{B}}{\mathrm{b}}+\frac{2 \cos \mathrm{C}}{\mathrm{c}}=\frac{\mathrm{a}}{\mathrm{bc}}+\frac{\mathrm{b}}{\mathrm{ca}}$ then $\angle \mathrm{A}=$

In a triangle ABC with usual notations, if $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in arithmetic progression, then, $\tan \frac{\mathrm{A}}{2} \cdot \tan \frac{\mathrm{C}}{2}=$

With usual notations, in a triangle $A B C$, if $\theta$ is any real number, then $a \cos (B-\theta)+b \cos (A+\theta)$ is

If two sides of a triangle are $\sqrt{3}-2$ and $\sqrt{3}+2$ units and their included angle is $60^{\circ}$, then the third side of the triangle is

In a triangle $\mathrm{ABC}, l(\mathrm{AB})=\sqrt{23}$ units, $l(\mathrm{BC})=3$ units, $l(\mathrm{CA})=4$ units, then $\frac{\cot A+\cot C}{\cot B}$ is

In a triangle ABC , with usual notations, $2 \mathrm{ac} \sin \left(\frac{\mathrm{A}-\mathrm{B}+\mathrm{C}}{2}\right)$ is equal to

In a triangle ABC , with usual notations, $\frac{\cos \mathrm{B}+\cos \mathrm{C}}{\mathrm{b}+\mathrm{c}}+\frac{\cos \mathrm{A}}{\mathrm{a}}$ has the value

The angles of a triangle are in the ratio $5: 1: 6$, then ratio of the smallest side to the greatest side is

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

If the sides of a triangle $a, b, c$ are in A.P., then with usual notations, a $\cos ^2 \frac{\mathrm{C}}{2}+\mathrm{c} \cos ^2 \frac{\mathrm{~A}}{2}$ is

If in a triangle $A B C$, with usual notations, the angles are in A.P. and $b: c=\sqrt{3}: \sqrt{2}$, then angle $\mathrm{A}=$

With usual notations, if the lengths of the sides of a triangle are $7 \mathrm{~cm}, 4 \sqrt{3} \mathrm{~cm}$ and $\sqrt{13} \mathrm{~cm}$, then the measures of the smallest angle is

In a triangle ABC , with usual notations, if $\mathrm{m} \angle \mathrm{A}=45^{\circ}, \mathrm{m} \angle B=75^{\circ}$, then $\mathrm{a}+\mathrm{c} \sqrt{2}$ has the

If the angles $\mathrm{A}, \mathrm{B}$ and C of a triangle ABC are in the ratio $2: 3: 7$ respectively, then the sides a, b and c are respectively in the ratio

If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are the angles of a triangle with $\tan \frac{A}{2}=\frac{1}{3}, \tan \frac{B}{2}=\frac{2}{3}$ then the value of $\tan \frac{C}{2}$ is

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

For the triangle ABC , with usual notations, if the angles $A, B, C$ are in A.P. and $\mathrm{m} \angle \mathrm{A}=30^{\circ}, \mathrm{c}=3$, then the values of a and b are respectively

If $(a+b) \cos C+(b+c) \cos A+(c+a) \cos B=72$ and if $a=18, b=24$, then area of the triangle $A B C$ is

If the angles of a triangle are in the ratio $4: 1: 1$, then the ratio of the longest side to the perimeter is

In $\triangle \mathrm{ABC}$, with usual notations, if $\mathrm{b}=3$, $c=8, \mathrm{~m} \angle \mathrm{~A}=60^{\circ}$, then the circumradius of the triangle is _______ units.

If the lengths of the sides of triangle are 3,5,7, then the largest angle of the triangle is

In $\triangle A B C$, with usual notations, if $\frac{1}{b+c}+\frac{1}{c+a}=\frac{3}{a+b+c}$, then $m \angle C$ is equal to

In a triangle $$\mathrm{A B C, m \angle A, m \angle B, m \angle C}$$ are in A.P. and lengths of two larger sides are 10 units, 9 units respectively, then the length (in units) of the third side is

In $$\triangle \mathrm{ABC}$$, with usual notations, $$2 \mathrm{ac} \sin \left(\frac{1}{2}(\mathrm{~A}-\mathrm{B}+\mathrm{C})\right)$$ is equal to

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

In $$\triangle A B C$$, with usual notations, if $$\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$$, then the value of $$\cos A+\cos B+\cos C$$ is

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one, then the sides of the triangle (in units) are

In $$\triangle A B C$$ with usual notation, $$\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$$ and $$a=\frac{1}{\sqrt{6}}$$, then the area of triangle is _______ sq. units.

Angles of a triangle are in the ratio $$4: 1: 1$$. Then the ratio of its greatest side to its perimeter is

The lengths of sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the length of the sides of the triangle (in units) are

If two angles of $$\triangle \mathrm{ABC}$$ are $$\frac{\pi}{4}$$ and $$\frac{\pi}{3}$$, then the ratio of the smallest and greatest sides are

In $$\triangle \mathrm{ABC}, \mathrm{m} \angle \mathrm{B}=\frac{\pi}{3}$$ and $$\mathrm{m} \angle \mathrm{C}=\frac{\pi}{4}$$. Let point $$\mathrm{D}$$ divide $$\mathrm{BC}$$ internally in the ratio $$1: 3$$, then $$\frac{\sin (\angle B A D)}{\sin (\angle C A D)}$$ has the 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 $$x^2-\mathrm{c}^2=y$$, where $$\mathrm{c}$$ is the length of the third side of the triangle, then the circumradius of the triangle is

If the vertices of a triangle are $$(-2,3),(6,-1)$$ and $$(4,3)$$, then the co-ordinates of the circumcentre of the triangle are

In triangle $$\mathrm{ABC}$$ with usual notations $$\mathrm{b}=\sqrt{3}, \mathrm{c}=1, \mathrm{~m} \angle \mathrm{A}=30^{\circ}$$, then the largest angle of the triangle is

If the angles of a triangle are in the ratio $$4: 1: 1$$, then the ratio of the longest side to its perimeter is

If in $$\triangle \mathrm{ABC}$$, with usual notations, $$a \cdot \cos ^2 \frac{C}{2}+c \cos ^2 \frac{A}{2}=\frac{3 b}{2}$$, then

In a triangle $$\mathrm{ABC}$$, with usual notations, if $$\mathrm{m} \angle \mathrm{A}=60^{\circ}, \mathrm{b}=8, \mathrm{a}=6$$ and $$\mathrm{B}=\sin ^{-1} x$$, then $$x$$ has the value

In a triangle $$\mathrm{ABC}$$, with usual notations, if $$\mathrm{c}=4$$, then value of $$(a-b)^2 \cos ^2 \frac{C}{2}+(a+b)^2 \sin ^2 \frac{C}{2}$$ is

If one side of a triangle is double the other and the angles opposite to these sides differ by $$60^{\circ}$$, then the triangle is

In $$\triangle \mathrm{PQR}, \sin \mathrm{P}, \sin \mathrm{Q}$$ and $$\sin \mathrm{R}$$ are in A.P., then

Let $$a, b, c$$ be the lengths of sides of triangle $$A B C$$ such that $$\frac{a+b}{7}=\frac{b+c}{8}=\frac{c+a}{9}=k$$. Then $$\frac{(\mathrm{A}(\triangle \mathrm{ABC}))^2}{\mathrm{k}^4}=$$

In $$\triangle \mathrm{ABC}$$, with usual notations, $$\mathrm{m} \angle \mathrm{C}=\frac{\pi}{2}$$, if $$\tan \left(\frac{A}{2}\right)$$ and $$\tan \left(\frac{B}{2}\right)$$ are the roots of the equation $$a_1 x^2+b_1 x+c_1=0\left(a_1 \neq 0\right)$$, then

Two sides of a triangle are $$\sqrt{3}+1$$ and $$\sqrt{3}-1$$ and the included angle is $$60^{\circ}$$, then the difference of the remaining angles is

In a triangle ABC with usual notations a = 2, b = 3, then value of $$\frac{\cos 2 \mathrm{~A}}{\mathrm{a}^2}-\frac{\cos 2 \mathrm{~B}}{\mathrm{~b}^2}$$ is

In any $$\triangle A B C$$, with usual notations, $$c(a \cos B-b \cos A)=$$

If in a $$\triangle A B C$$, with usual notations, $$\mathrm{a}^2, \mathrm{~b}^2, \mathrm{c}^2$$ are in A.P. then $$\frac{\sin 3 B}{\sin B}=$$

If $$\mathrm{G}(\overline{\mathrm{g}}), \mathrm{H}(\overline{\mathrm{h}})$$ and $$\mathrm{P}(\overline{\mathrm{p}})$$ are respectively centroid, orthocenter and circumcentre of a triangle and $$\mathrm{x} \overline{\mathrm{p}}+\mathrm{y} \overline{\mathrm{h}}+z \overline{\mathrm{g}}=\overline{0}$$, then $$\mathrm{x}, \mathrm{y}, \mathrm{z}$$ are respectively.

With usual notations in $$\triangle$$ABC, if $$\frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)}$$, then $$a^2, b^2, c^2$$ are in

The area of the triangle $$\mathrm{ABC}$$ is $$10 \sqrt{3} \mathrm{~cm}^2$$, angle $$\mathrm{B}$$ is $$60^{\circ}$$ and its perimeter is $$20 \mathrm{~cm}$$, then $$\ell(\mathrm{AC})=$$

In a triangle $$\mathrm{ABC}$$, with usual notations $$\mathrm{a}=2, \mathrm{~b}=3, \mathrm{c}=5$$, then $$\frac{\cos \mathrm{A}}{\mathrm{a}}+\frac{\cos \mathrm{B}}{\mathrm{b}}+\frac{\cos \mathrm{C}}{\mathrm{c}}=$$

In $$\Delta ABC$$, with usual notations $$\mathrm{\frac{b\sin B-c\sin C}{\sin(B-C)}}=$$

With usual notations, in any $$\triangle A B C$$, if $$a\cos B=b \cos A$$, then the triangle is

In $$\triangle A B C$$, with usual notations, $$2 a b \sin \frac{1}{2}(A+B-C)=$$

If in $$\Delta$$ABC, with usual notations, the angles are in A.P., then $$\mathrm{\frac{a}{c}}$$ sin 2 C + $$\mathrm{\frac{c}{a}}$$ sin 2 A =

With usual notations, perimeter of a triangle $$A B C$$ is 6 times the arithmetic mean of sine of its angles. If $$\mathrm{a}=1$$, then measure of angle $$\mathrm{A}=$$

With usual notations if the angles of a triangle are in the ratio 1 : 2 : 3, then their corresponding sides are in the ratio.

With usual notations, if the angles $A, B, C$ of a $\triangle A B C$ are in $A P$ and $b: c=\sqrt{3}: \sqrt{2}$

The area of the $\triangle A B C$ is $10 \sqrt{3} \mathrm{~cm}^2$, angle $B$ is $60^{\circ}$ and its perimeter is 20 cm , then $\ell(A C)=$

In a $$\triangle A B C$$ if $$2 \cos C=\sin B \cdot \operatorname{cosec} A$$, then

In a triangle $$A B C$$ with usual notations, if $$\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$$, then area of triangle $$A B C$$ with $$a=\sqrt{6}$$ is

In a triangle $$A B C$$, if $$\frac{\sin A-\sin C}{\cos C-\cos A}=\cot B$$, then $$A, B, C$$, are in

In $\triangle A B C$, with the usual notations, if $\left(\tan \frac{A}{2}\right)\left(\tan \frac{B}{2}\right)=\frac{3}{4}$ then $a+b=\ldots \ldots$

In $\triangle A B C$, with the usual notations, if $\sin B \sin C=\frac{b c}{a^2}$, then the triangle is. ...........

If $R$ is the circum radius of $\triangle A B C$, then $A(\triangle A B C)=\ldots \ldots$

In $\triangle A B C$, if $\tan A+\tan B+\tan C=6$ and $\tan A \cdot \tan B=2$ then $\tan C=$ ...........

In $\triangle A B C$; with usual notations, $$\frac{b \sin B-c \sin C}{\sin (B-C)}=\ldots \ldots$$

In $\triangle A B C$; with usual notations, if $\cos A=\frac{\sin B}{\sin C}$ then the triangle is ............