Question

Read the following passage:

Jagdish has a field which is in the shape of a right angled triangle AQC. He wants to leave a space in the form of a square PQRS inside the field for growing wheat and the remaining for growing vegetables (as shown in the figure). In the field, there is a pole marked as O.

Based on the above information, answer the following questions :

1. Taking O as origin, coordinates of P are (–200, 0) and of Q are (200, 0). PQRS being a square, what are the coordinates of R and S?
2. (a) What is the area of square PQRS?
   OR
   (b) What is the length of diagonal PR in square PQRS?
3. If S divides CA in the ratio K : 1, what is the value of K, where point A is (200, 800)?

Answer 1

i. Since, PQRS is a square

∴ PQ = QR = RS = PS

Length of PQ = 200 – (–200) = 400

∴ The coordinates of R = (200, 400)

and coordinates of S = (–200, 400)

ii. (a) Area of square PQRS = (side) 2

= (PQ)²

= (400)²

= 1,60,000 sq. units

OR

(b) By Pythagoras theorem

(PR)² = (PQ)² + (QR)²

= 1,60,000 + 1,60,000

= 3,20,000

`\implies` PR = `sqrt(3,20,000)`

= `400 xx sqrt(2)` units

iii. Since, point S divides CA in the ratio K : 1

∴ `((Kx_2 + x_1)/(K + 1), (Ky_2 + y_1)/(K + 1))` = (–200, 400)

`implies ((K(200) + (-600))/(K + 1), (K(800) + 0)/(K + 1))` = (–200, 400)

`implies ((200K - 600)/(K + 1), (800K)/(K + 1))` = (–200, 400)

∴ `(800K)/(K + 1)` = 400

`implies` 800K = 400K + 400

`implies` 400K = 400

`implies` K = 1