Distance and Displacement

If the circumcentre of a triangle lie on the side whose adjacent angles are $45°$ each, then find the other two sides if the radius of the circumcircle is $15 \mathrm{cm}$?

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.

A man walks $4$ km. to East then he turns left walks $4$ km. and turns left and walk $4$ km. In which direction is he facing now?

If the diagonals of a rhombus are $8 \mathrm{units}$ and $6 \mathrm{units}$, then the square of its side is ;

A ladder $20 \mathrm{m}$ long is placed in street so as to reach a window $16 \mathrm{m}$ high. On turning the ladder on the other side of the street, it reaches a point $12 \mathrm{m}$ high.The width of the street is ; 

In a right-angled $∆ABC$, right angle at $A$, if $AD\perp BC$ such that $AD=p$. If $BC=a$, $CA=b$ and $AB=c$, then which relation is true?

Read the following information for answering the questions that follow:
On a playing ground M, N, Q, R and T are standing as described below facing the North.
N is $30$ metres to the right of R. M is $70$ metres to the South of N. Q is $10$ metres to the West of R. T is $100$ metres to the North of M.

If a boy walks from Q, meets R followed by N, M and then T, how many metres has he walked if he has travelled the straight distance all through?

Ram goes $10$ km in the North from his school. Now, turning to the left, he goes to $8$ km and again turns to left and goes to $10$ km. Find how far is he from his school and in which direction?

Arjun traveled 10 km Southward, then turned right and traveled 12 km, then turned right and traveled 10 km. How far was Arjun from the starting point?

Rohit walks 20 km towards the South. He turns left and walks another 10 km. He turns left and walks another 20 km, then turns in his right direction & walks 5 km. In which direction is he and at what distance from the starting point?

Read the following information and answer the following question:  

A man is facing South at point A. From there, he turns right and walks $20 \mathrm{m}$, then, he turns right and walks $25 \mathrm{m}$. Then, he turns left and walks $30 \mathrm{m}$ and after turning towards the north, walks $\mathrm{x} \mathrm{m}$. Then, he turns right again and walks $70 \mathrm{m}$ and reaches point B. If the angular position of point B with respect to the point A along vertical axis is $26.5^{\mathrm{o}}$, then find the value of $\mathrm{x}$ (in $\mathrm{m}$).

Read the following information and answer the following question:

A lady is facing west at point $\mathrm{A}$. From there, he turns right and walks $30 \mathrm{m}$ to the point $\mathrm{D}$, then, he turns right and walks $\mathrm{x} \mathrm{m}$. Then, he turns right again and walks $60 \mathrm{m}$ and reaches point $\mathrm{B}$. If the angular position of point $\mathrm{B}$ with respect to the point $\mathrm{A}$ along the horizontal axis is ${45}^{\mathrm{o}}$, then find the value of $\mathrm{x}$.

Read the following information and answer the following question:  

A lady is facing east at point $A$. From there, he turns left and walks $30 \mathrm{m}$, then, he turns right and walks $30 \mathrm{m}$. Then, he turns right again and walks $60 \mathrm{m}$ and reaches point $B$.

What is the angular position of point $B$ with respect to the point $A$ along the horizontal axis?

Read the following information and answer the following question:  

A man is facing South at point $A$. From there, he turns right and walks $40 \mathrm{m}$, then, he turns right and walks $40 \mathrm{m}$. Then, he turns right again and walks $80 \mathrm{m}$ and reaches point $B$.

What is the angular position of point $B$ with respect to the point $A$ along vertical axis?

Starting from the point X, Jaswant walked 10m towards west. He turned left and walked 15m. He then turned left and walked 15 m. After this he turned to his right and walked 20m. In which directions is now Jaswant from X?

Read the following information and answer the following question:  

A man is facing North at point $A$. From there, he turns left and walks $20 \mathrm{m}$, then, he turns right and walks $25 \mathrm{m}$. Then, he turns left and walks $30 \mathrm{m}$ and after turning towards the north, walks $15 \mathrm{m}$. Then, he turns right again and walks $70 \mathrm{m}$ and reaches point $B$.

What is the angular position of point $B$ with respect to the point $A$ along vertical axis?

Rishi walked $10\mathrm{m}$, towards north. Then he turned right and walks $15\mathrm{m}$. Then he turns right and walks $25\mathrm{m}$. Then, he turns left and walks $8\mathrm{m}$. Finally, he turns left and walks $20\mathrm{m}$. In which direction and how many metres is he from the starting position?

A woman walks $7\mathrm{km}$, towards south and then turns to the right. After walking $10\mathrm{km}$, she turns to the left and walks $8\mathrm{km}$. Now in which direction is she from the starting place?

A circular ring of diameter $4$ units is kept on $\mathrm{X}-\mathrm{Y}$ axis such that its centre coincides with the origin. Now, another circular ring of diameter which is half of the diameter of the former ring is rolled on the $\mathrm{X}$ axis from negative $\mathrm{X}$ axis to positive X axis passing over the bigger ring (when the bigger ring becomes the obstacle for the smaller one while rolling). What is the length (in units) of the part of the $\mathrm{X}$ axis which does not come in contact with the smaller ring?

Note: The smaller ring moves only in $\mathrm{X}-\mathrm{Y}$ plane.

A ladder leaning against a wall makes an angle $\theta$ with the horizontal ground such that $\cos \left(\theta \right)=\frac{5}{13}$. If the height of the top of the ladder from the wall is $18 \mathrm{m}$, then what is the distance (in $\mathrm{m}$) of the foot of the ladder from the wall?