Question

The figure given below shows a school ground OABC such that \(O A=10 \mathrm{~m}\) and \(OC=100 \mathrm{~m}\). On a founders day, some sports activities were performed on this ground. Ten vertical lines have been drawn with chalk powder at intervals of 1 m each, as shown below. Geeta runs \(\frac{1}{4}\) of the distance OC on the 2nd line and posts a white flag W. Reena runs \(\frac{1}{5}\) of the distance OC on the 8th line and posts a red flag R.

Look at the given figure and answer the following questions.

1. What are the coordinates of the point W?
2. What are the coordinates of the point R?
3. What is the distance between the points W and R?
4. What are the coordinates of the point which is exactly halfway between the line segment joining W and R?
5. What are the coordinates of the point which is at about one fourth distance in the line segment joining W and R?

Answer 1

i. Coordinates of point W:

Geeta runs on the 2nd line, so the x-coordinate is 2.

Distance along the y-axis \((OC = 100\text{ m})\): \[y_W = \frac{1}{4} \times 100 = 25\text{ m}\]

\[(2, 25)\]

ii. Coordinates of point R:

Reena runs on the 8th line, so the x-coordinate is 8.

Distance along the y-axis \((OC = 100\text{ m})\): \[y_R = \frac{1}{5} \times 100 = 20\text{ m}\]

\[(8, 20)\]

iii. Distance between points W and R:

Formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Substitution: \[WR = \sqrt{(8 - 2)^2 + (20 - 25)^2}\]

\[WR = \sqrt{6^2 + (-5)^2}\]

\[= \sqrt{36 + 25}\]

\[= \sqrt{61}\text{ m}\]

iv. Coordinates of the halfway point Midpoint of WR:

Midpoint Formula: \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]

Calculation: \[M = \left(\frac{2 + 8}{2}, \frac{25 + 20}{2}\right)\]

\[= \left(\frac{10}{2}, \frac{45}{2}\right)\]

\[= (5, 22.5)\]

v. Coordinates of the point at one-fourth distance from W to R:

The point at one-fourth distance from W to R is the midpoint of the line segment joining \(W(2, 25)\) and the midpoint \(M(5, 22.5)\):

\[P = \left(\frac{2 + 5}{2}, \frac{25 + 22.5}{2}\right)\]

\[= \left(\frac{7}{2}, \frac{47.5}{2}\right)\]

\[= (3.5, 23.75)\]