Heights And Distances

At a point $P$ on the ground, the angle of elevation of the top of a $10\mathrm{m}$ tall building and of a helicopter hovering some distance over the top of the building are $30º$ and $60º$ respectively. Then, the height of the helicopter above the ground is

At a point $A$, the angle of elevation of a tower is found to be such that its tangent is $\frac{ 5}{12}$ . On walking $192 \mathrm{meter}$ towards the tower, the tangent of the angle of elevation is found to be $\frac{3}{4}$ . Find the height of the tower.the

The angles of depression of Two ships from the top of a lighthouse are $45°$ and $30°$ towards east, if the ships are $200 \mathrm{m}$ apart, the height of lighthouse is$?$ (Take $\sqrt{3}= 1.73$ )

A point $D$ is taken from the side $BC$ of a right-angled triangle $∆\mathrm{ABC}$, where $AB$ is the hypotenuse. Find the appropriate relation.

If $D$ and $E$ are points on the sides $AB$ and $AC$ respectively of a $∆ABC$ such that $DE\parallel BC$. If $AD = x$, $DB = x - 2$, $AE = x + 2$ and $EC = x - 1$, then find the value of $x$.

Which is not correct?

A telegraph post gets broken at a point against a storm and its top touches the ground at a distance $20 \mathrm{m}$ from the base of the post making an angle ${30}^{°}$ with the ground. What is the height of the post (in $\mathrm{m}$)?

A man is watching form the top of the tower a boat speeding away from the tower. The boat makes the angle of depression of $45°$ with the man's eye when at a distance of$60 \mathrm{m}$ from the tower. After $5 \sec$ the angle of depression becomes $30°$. What is the approximate speed of the boat, assuming that it is running in still water ?

A man is watching from the top of a tower a boat speeding away from the tower. The boat makes an angle of depression of ${45}^{°}$ with the man's eye when at a distance of $60 \mathrm{m}$ from the bottom of tower. After $5 \mathrm{seconds}$, the angle of depression becomes ${30}^{°}$. What is the approximate speed of the boat assuming that it is running in still water?

The sides of a triangle are $6\mathrm{cm}, 11\mathrm{cm}$ and $15\mathrm{cm}.$ The radius of its incircle is:

The angles of elevation of the top of a tower from the top and the foot of a pole of height $10\mathrm{m}$ are $30°$ and $60°$ respectively. Then the height of the tower is...-

$\mathrm{ABCD}$ is a square. The diagonals $\mathrm{AC} \mathrm{and} \mathrm{BD}$ meet at $\mathrm{O}$ . Let $\mathrm{K}, \mathrm{L}$ be the points on $\mathrm{AB}$ such that $\mathrm{AO} = \mathrm{AK}$, $\mathrm{BO}= \mathrm{BL}$. If $\angle \mathrm{LOK}=\mathrm{\theta }$, then what is the value of $\tan \mathrm{\theta }?$

Two boats leave a place at the same time. One travels $56 \mathrm{km}$ in the direction $\mathrm{N} 40° \mathrm{E}$, while the other travels $48 \mathrm{km}$ in the direction $\mathrm{S} 80° \mathrm{E}$. What is the distance between the boats?

The angle of depression of a vehicle on the ground from the top of a tower is $60°$. If the vehicle is at a distance of $100 \mathrm{meters}$ away from the building, find the height of the tower.

If $sin\theta +cos\theta =1$ , then $sin\theta cos\theta$  is equal to:

A straight tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of ${30}^{°}$ with the ground. The distance from the foot of the tree of the point where the top touches the ground is $10 \mathrm{m}$. The height of the tree (in $\mathrm{m}$) is ________.

Find the value of ${\cos }^6\mathrm{X}+{\sin }^6\mathrm{X}-1+3{\sin }^2\mathrm{X} {\cos }^2\mathrm{X}$ $=?$

The angle of depression of a point on the ground as seen from the top of a tower, $50 \mathrm{ft}$ high, is $45°$. Find the distance of the point on the ground from the foot of the tower.

The angle of elevation from the end of the shadow to the top of the building is $56°$ and the distance is $120\mathrm{ft}$. What is the length of the shadow$?$

If the ratio of sines of angles of a triangle is $1:1:\sqrt{2}$ , then the ratio of squares of the greatest side to sum of the squares of other two sides is _____.