Properties of triangles practice · Mixed difficulty

The angle of depression of a point situated at a distance of $70 \mathrm{~m}$ from the base of a tower is $45^{\circ}$. The height of the tower is

The perimeter of a $\Delta \;ABC$ is 6 times the arithmetic mean of the sine ratios of its angles. If $a = 1$, then $A$ is equal to

The value of $\frac{1}{{r_1^2}} + \frac{1}{{r_2^2}} + \frac{1}{{r_3^2}} + \frac{1}{{{r^2}}}$ 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

In a $\Delta\; ABC,\;a = 5,\;b = 7$ and $\sin A = \frac{3}{4}$, then the number of possible triangles are

In a $\Delta \;ABC,\;\mathrm{cosec}\;A\left( {\sin B\cos C + \cos B\sin C} \right)$ is equal to

If $AD,\,\,BE$ and $CF$ are the medians of a $\Delta \,ABC$, then $\left( {A{D^2} + B{E^2} + C{F^2}} \right)\;:\,\,\left( {B{C^2} + C{A^2} + A{B^2}} \right)$ is equal to

In an obtuse angle triangle $A B C, a=\sqrt{3}+1, B=\frac{\pi}{3}$ and $R=\sqrt{2}$, then angle $C$ is

In $\Delta \;ABC,\angle A = \frac{\pi }{2},b = 4,c = 3$, then the value of $\frac{R}{r}$ is equal to

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

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?

If $r_1, r_2, r_3$ are ex-radii of the encircles of triangle $A B C$ then $(s-a) r_1+(s-b) r_2+(s-c) r_3$ is

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 $\triangle A B C, \frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}+\frac{1}{r^2}$ is equal to

If in $\Delta \;ABC,\sin \frac{A}{2}\sin \frac{C}{2} = \sin \frac{B}{2}$ and $2s$ is the perimeter of the triangle, then s is

If $c^2=a^2+b^2$, then $4 s(s-a)(s-b)(s-c)$ is equal to

If in a $\Delta \;ABC\;a = 5,\;b = 4,\;A = \frac{\pi }{2} + B$, then $C$

If in a $\Delta \,ABC,\,\,3\,\,\sin A = 6\,\,\sin B = 2\,\sqrt 3 \,\,\sin C$ then angle $A$ is

If the area of a triangle $ABC$ is $\Delta$, then ${a^2}\sin 2B + {b^2}\sin 2A$ is equal to

Let $p, q$ and $r$ be the sides opposite to the angles $P, Q$ and $R$, respectively in a $\triangle P Q R$. If $r^2 \sin P \sin Q=p q$, then the triangle is

$a^3 \cos (B-C)+b^3 \cos (C-A)+c^3 \cos (A-B)$ is equal to

In a $\Delta ABC,\mathrm{let}\angle C = \frac{\pi }{2}$, if $r$ is the inradius and $R$ is the circumradius of the $\Delta \;ABC$, then $2\left( {r + R} \right)$ equals

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

In a triangle, the lengths of the two larger sides are 10 cm and 9 cm respectively. If the angles of the triangle are in AP, then the length of the third side in cm can be

The angle of elevation of the top of a vertical tower from a point $\mathrm{A}$, due east of it is $45^{\circ}$. The angle of elevation of the top of the same tower from a point $B$, due south of $A$ is $30^{\circ}$. If the distance between $A$ and $B$ is $54 \sqrt{2} \mathrm{~m}$, then the height of the tower (in metres), is :

If the sides of a triangle are in ratio 3:7:8, then $R:r$ is equal to

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