Question 3

Coordinate Geometry in Real Life: Coordinate geometry lets us represent physical locations as points on a grid using $(x, y)$ coordinates. The x-axis and y-axis act like the rows and columns of the school ground. Once locations are expressed as coordinates, we can use mathematical formulas to find distances, midpoints, and more — without physically measuring.

Formulas used in this solution:

Distance between two points $\text{A}(x_1, y_1)$ and $\text{B}(x_2, y_2)$:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Gives the straight-line length between any two locations on the grid.

Midpoint of a line segment joining $\text{A}(x_1, y_1)$ and $\text{B}(x_2, y_2)$:

$M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$

Gives the exact centre point between two locations.

Section Formula — coordinates of point $\text{P}$ dividing $\text{AB}$ in ratio $m:n$ internally:

$x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}$

Gives the location of a point that splits a path into two specific proportions.

We will use coordinate geometry to find distances and midpoints.

Step 1 — Find Niharika's flag coordinates

Niharika runs on the 2nd line. Her x-coordinate is 2. She runs $\frac{1}{4}$th the distance AD. The total distance AD is 100m.

$y_{\text{Niharika}} = \frac{1}{4} \times 100$

$= 25$

$\boxed{\text{Niharika's flag is at } (2, 25)}$

Step 2 — Find Preet's flag coordinates

Preet runs on the 8th line. Her x-coordinate is 8. She runs $\frac{1}{5}$th the distance AD. The total distance AD is 100m.

$y_{\text{Preet}} = \frac{1}{5} \times 100$

$= 20$

$\boxed{\text{Preet's flag is at } (8, 20)}$

Step 3 — Calculate distance between flags

Let Niharika's flag be $N(x_1, y_1) = (2, 25)$. Let Preet's flag be $P(x_2, y_2) = (8, 20)$. We use the distance formula.

$\text{Distance NP} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

$= \sqrt{(8 - 2)^2 + (20 - 25)^2}$

$= \sqrt{(6)^2 + (-5)^2}$

$= \sqrt{36 + 25}$

$= \sqrt{61}$

$\boxed{\text{Distance between flags} = \sqrt{61} \text{ meters}}$

Step 4 — Find Rashmi's flag position

Rashmi posts her flag exactly halfway. We use the midpoint formula. Let $M(x, y)$ be the midpoint.

$x = \frac{x_1 + x_2}{2}$

$x = \frac{2 + 8}{2}$

$x = \frac{10}{2}$

$x = 5$

$y = \frac{y_1 + y_2}{2}$

$y = \frac{25 + 20}{2}$

$y = \frac{45}{2}$

$y = 22.5$

$\boxed{\text{Rashmi's flag is at } (5, 22.5)}$

Answer

(i) The distance between both flags is $\sqrt{61}$ meters. (ii) Rashmi should post her flag on the 5th line at a distance of 22.5m from AD.