Amit M Agarwal Solutions for Chapter: Properties of Triangles, Heights and Distances, Exercise 4: Target Exercises

The angles of elevation of the top of a tower from two points (collinear with foot of tower) on the ground at a distance $a$ and $b\left(b>a\right)$ from its foot are found to be $\alpha$ and $\beta .$ If $h$ is the height of tower, then

A flagstaff standing vertically at the vertex $A$ of an isosceles $\mathrm{\Delta }ABC,$ subtends angle $\frac{\pi }{3}$ at the mid-point of the base $BC$ and an angle $\frac{\pi }{6}$ at the vertex $B$. If $BC=a$, then height of flagstaff is

A tower of height $h$ standing vertically at the centre of a square of side length $\mathrm{a}$ subtends the same angle $\theta$ at all the corner points of the square. Then,

A monument $ABCD$ stands at a point $A$ on a level ground. At a point $P$ on the ground, the portions $AB, AC, AD$ subtend angles $\alpha , \beta , \gamma$, respectively. If $AB=a, AC=b, AD=c, AP=x$ and $\alpha +\beta +\gamma =180°,$ then

Three vertical tower standing at $A,B,C$ subtends the angle ${\theta }_A,{\theta }_B$ and ${\theta }_{\mathrm{C}},$ respectively at the circumcentre of $\mathrm{\Delta }ABC,$ then $\tan {\theta }_A,\tan {\theta }_B$ and $\tan {\mathrm{\theta }}_{\mathrm{C}}$ are in

On the top of a hemispherical dome of radius $r$, there stands a flag of height $h .$ From a point on the ground the elevation of the top of the flag is $30°.$ After moving a distance $d$ towards the dome, when the flag is just visible, then elevation is $45°,$ then $r$ and $h$ are

A spherical ball of radius $r$ subtends an angle ${\mathrm{\theta }}_1$ at a point $P$ on the ground. If the angle of elevation of centre of the ball at $P$ is ${\mathrm{\theta }}_2,$ then height of the centre of the ball from the ground is

A circular ring of radius $r \mathrm{m}$ is suspended from a point $h \mathrm{m}$ vertically above the centre of ring by four strings of equal lengths attached at equal intervals to the circumference of the ring. If $\theta$ is the angle between two consecutive strings, then $\cos \theta$ is equal to