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Question
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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.
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Look at the given figure and answer the following questions.
- What are the coordinates of the point W?
- What are the coordinates of the point R?
- What is the distance between the points W and R?
- What are the coordinates of the point which is exactly halfway between the line segment joining W and R?
- What are the coordinates of the point which is at about one fourth distance in the line segment joining W and R?
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Solution
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 \(M(5, 22.5)\):
\[P = \left(\frac{2 + 5}{2}, \frac{25 + 22.5}{2}\right)\]
\[= \left(\frac{7}{2}, \frac{47.5}{2}\right)\]
\[= (3.5, 23.75)\]
