Properties of triangles practice · Medium

The area of the triangle whose sides are $6,\;5,\;\sqrt {13}$ (in square unit) is

In a right-angled triangle, the sides are $a,\;b$ and $c$, with $c$ as hypotenuse, and $c - b \ne 1,\;c + b \ne 1$. Then the value of $\left( {{{\log }_{c + b}}a + {{\log }_{c - b}}a} \right)/\left( {2{{\log }_{c + b}}a \times {{\log }_{c - b}}a} \right)$ will be

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

If in a $\triangle A B C, \angle B=90^{\circ}$, then $\tan ^2\left(\frac{A}{2}\right)$ is equal to

If $A,\;B,\;C,\;D$ are the angles of a quadrilateral, then $\frac{{\sum \tan A}}{{\sum \cot A}}$ is equal to

In a $\triangle A B C, a, b, A$ are given and $b_1, b_2$ are two values of the third side $b$ such that $b_2=2 b_1$. Then, $\sin A$ is equal to

In a $\Delta \;ABC$, angle $A$ is greater than angle $B$. If the measures of angles $A$ and $B$ satisfy the equation $3\sin x - 4{\sin ^3}x - k = 0,\;\;0 < k < 1$, then the measure of angle $C$ is

In a triangle the length of the two larger sides are $\mathrm{24}$ and $\mathrm{22}$, respectively. If the angles are in AP, then the third side is

$A B C$ is an equilateral triangle of each side a $(>0)$. The inradius of the triangle is

If the angles of a right angled triangle are in A.P., then the ratio of the in-radius and the perimeter is

In $\triangle A B C, a=5, b=4$ and $\cos (A-B)=\frac{31}{32}$, then side $c$ is

Let $a, b$ and $c$ be the sides of a triangle and $\frac{a^2+b^2+c^2}{a b+b c+c a}=P$. Then

If in $\triangle A B C, \angle A=90^{\circ}$ and $c, \sin B, \cos B$ are rational numbers, then

If the ex-radii of a triangle are in H.P., then the corresponding sides are in

In any $\triangle A B C$, if $C=90^{\circ}$, then $\tan \frac{B}{2}$ is equal to

In a triangle $\mathrm{ABC}, 2 a^2+4 b^2+c^2=4 a b+2 a c$, then $\cos \mathrm{B}$ is equal to

In $\Delta \;ABC,{a^2}\left( {{{\cos }^2}B - {{\cos }^2}C} \right) + {b^2}\left( {{{\cos }^2}C - {{\cos }^2}A} \right) + {c^2}\left( {{{\cos }^2}A - {{\cos }^2}B} \right)$ is equal to

In a $\Delta \,ABC$, angles $A,\,\,B,\,\,C$ are in A.P., then
$\lim\limits_{A \to C} \,\,\frac{{\sqrt {3 - 4\,\,\sin \,\,A\,\,\sin C} }}{{\left| {\,A - C\,} \right|}}$ is equal to

The angles of elevation of a vertical tower standing inside a triangular field at the vertices of the field are each equal to $\theta$. If the length of the sides of the field are $30 \mathrm{~m}, 50 \mathrm{~m}$ and $70 \mathrm{~m}$, the height of the tower is

In a $\triangle \mathrm{ABC}, \mathrm{A}: \mathrm{B}: \mathrm{C}=3: 5: 4$.Then $\mathrm{a}+\mathrm{b}+\mathrm{c} \sqrt{2}$ is equal to

The sines of two angles of a triangle are equal to $5 / 13$ and $99 / 101$. The cosine of the third angle is

A triangle be such that $\cos 2 A+\cos 2 B+\cos 2 C$ is minimum. If in radius of the triangle is 3, then which of the following is CORRECT?

In any $\triangle A B C$, if $\sin A, \sin B$ and $\sin C$ are in $A P$, then the maximum value of $\tan \frac{B}{2}$ is

If $k$ is the perimeter of $\triangle A B C$, then $b \cos ^2 \frac{C}{2}+c \cos ^2 \frac{B}{2}$ is equal to

If two forces acting at right angles have their resultant $\sqrt{10} {~kg}$. wt. and when they act at an angle of $60^{\circ}$, the resultant is $\sqrt{13} {~kg}$. wt., then the forces are

In any triangle $A B C$, if $\cos A=\frac{\sin B}{2 \sin C}$, then

If in a $\Delta \,ABC,\,\,A = \frac{\pi }{3}$ and $AD$ is the median, then

If ${A_1}\,{A_2}\,{A_3}\,{A_4}\,{A_5}$ be a regular pentagon inscribed in a unit circle. Then $\left( {{A_1}{A_2}} \right)\,\,\left( {{A_1}{A_3}} \right)$ is equal to

If $a, b$ and $A$ are given in $a$ triangle and $c_1, c_2$ are the possible values of the third side, then $c_1^2+c_2^2-2 c_1 c_2 \cos 2 A$ is equal to

If in a $\triangle A B C, a^2+b^2+c^2=8 R^2$, where $R=$ circumradius, the the triangle is