Rohan and Sohan stay in a village which are 50 km apart and are situated on Delhi Agra Highway as shown in the figure. Another highway YY' crosses Agra Delhi Highway at O (0, 0). A small local road PQ crosses both the highways at points A and B such that OA = 10 km and OB = 12 km. Also the villages of Sita and Gita are on the smaller highway YY'. Sita's village is 12 km from O and Gita's village is 15 km from O.

Now answer the following questions:

(i) Find the Coordinates of A and B

(ii) Find the lengths of local road AB.

(iii) What is the shortest distance Sohan will cover to reach Sita's village.

(iv) How far is Rohan's village from Gita's village ?

(i) Point A is on the X-axis at a distance of 10 km from O (0, 0). So, A = (10, 0).

Point B is on the Y-axis at a distance of 12 km from O (0, 0). So, B = (0, 12).

Hence, coordinates of A = (10, 0) and coordinates of B = (0, 12).

(ii) Using Pythagoras theorem for the △OAB:

AB² = BO² + OA²

AB² = 12² + 10²

AB² = 144 + 100

AB² = 244

AB = $\sqrt{244}$

AB = 15.6 km.

Hence, length of the local road AB = 15.6 km.

(iii) Based on the diagram, Sohan's village is at (30, 0) and Sita's village is at (0, 12).

$\text{Distance} = \sqrt{(x$2 - x1)^2 + (y2 - y1)^2} \\[1em] = \sqrt{(30 - 0)^2 + (0 - 12)^2} \\[1em] = \sqrt{30^2 + (-12)^2} \\[1em] = \sqrt{900 + 144} \\[1em] = \sqrt{1044} \\[1em] = 32.31 \text{ km}

Hence, shortest distance between Sohan's village and Sita's village = 32.31 km.

(iv) Distance of Rohan's village from Gita's village:

Distance between Rohan's village to O + Distance between Gita's village to O

= 20 + 15

= 35 km.

Hence, distance from Rohan's village to Gita's village = 35 km.