Question

(c) On Annual Sports Day of a school, lines have been drawn with lime powder at a distance of 1 m each in the rectangular shaped school playground. 80 plastic cones have been placed at a distance of 1 m from each other along PY as shown in figure. Pushpendra runs th the distance PY on the 3rd line and post a yellow flag. Pankaj runs th the distance PY on the 7th line and post a blue flag.



Based on the above information, answer the following questions:

(i) Find the coordinates of the yellow flag and blue flag.

(ii) What is the distance between both the flags?

(iii) If Raman has to post a green flag exactly halfway between the line segments joining the two flags, where should he post his flag?

(iv) If Raman change his position and post a green flag at a point between the line segment, joining the two flags, then find the coordinate of the green flag which divides the line segment internally in the ratio 1: 2.​

Answer 1

Given the orientation of a sports field PXQY , answer the given question

Explanation:

1. let the rectangular playground be denoted by PXQY.
2. Since the lines are drawn at one meter apart and cones are also placed one meter apart from each other along PY we can consider the playground as x-y co-ordinate plane with PX and PY as x and y-axis respectively.
3. length of PY is , $PY=80(1)=80m$  
4. now Pushpendra runs $\frac{1}{4}^{th$ of PY hence his co-ordinates are,                           [tex]x_p=line\ number= 3\\ y_p=\frac{PY}{4}=\frac{80}{4} =20 \\ co-ordinates\ of\ yellow\ flag= P(3,20) [/tex]   ----ANS
5. now Pankaj runs $\frac{1}{5}^{th$ of PY hence his co-ordinates are,                                     $x_j=line\ number= 7\\y_j=\frac{PY}{5}=\frac{80}{5} =16 \\co-ordinates\ of\ blue\ flag= J(7,16)$   -----ANS
6. Distance between both the flags is the distance between points P and J,                                                                                                                 [tex]d=\sqrt{(x_p-x_j)^2+(y_p-y_j)^2} \\ d=\sqrt{(3-7)^2+(20-16)^2} \\ d=\sqrt{16+16}\\ d=4\sqrt{2}\ m [/tex]         -----ANS
7. now the co-ordinates of green flag are the mid-point of points P and J,                                                                                                         [tex]x_m=\frac{x_p+x_j}{2} \ \ \ \ \ \ \ y_m=\frac{y_p+y_j}{2} \\\\ x_m=\frac{3+7}{2} \ \ \ \ \ \ \ \ \ \ y_m=\frac{20+16}{2} \\\\ x_m=5,\ \ \ y_m=18\\ co-ordinate\ of\ green\ flag\ is\ M(5,18)[/tex]      ------ANS
8. now co-ordinates of green flag if it divides the line PJ in ratio $1:2$ is,                                                                                                              $x_g=\frac{(1)x_p+(2)x_j}{1+2} \ \ \ \ \ \ \ y_g=\frac{(1)y_p+(2)y_j}{1+2} \\\\x_g=\frac{3+14}{3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ y_g=\frac{20+32}{3} \\\\x_g=\frac{17}{3} ,\ \ \ y_g=\frac{52}{3} \\co-ordinate\ of\ green\ flag\ is\ M(5.7,17.3)\ approximately$   -------ANS