Question

Three friends Govind, Rishi and Kanika are participating in a Treasure Hunt organized in their school playground. The playground is mapped using a coordinate grid where each square represents 1 meter.

At a certain point in the game, they each stand at different spots waiting for their next clue. Their positions are recorded on the grid as points:

• Govind is at point P
• Rishi is at point Q
• Kanika is at point R 

The coordinate map is shown alongside. 

Based on the above information, answer the following questions:

1. Is Q the midpoint of segment PR? Justify your answer.
2. A new clue directs them to reach point M, which divides segment PQ in the ratio 2 : 3. Find the coordinates of M.

Answer 1

(i) From graph,

Coordinates of P = (4, 13), Q = (7, 8) and R = (11, 2)

By mid-point formula,

Mid-point = `((x_1 + x_2)/2,(y_1 + y_2)/2)`

Substituting values we get :

Mid-point of PR = `((4 + 11)/2, (13+2)/2)`

= `(15/2, 15/2)`

= (7.5, 7.5)

Hence, Q is not the midpoint of segment PR.

(ii) Given,

M divides PQ in the ratio 2 : 3.

By section formula,

M = `((m_1x_2 + m_2x_1)/(m_1 + m_2), (m_1y_2 + m_2y_1)/(m_1 + m_2))`

Substituting values we get:

⇒ M = `((2 × 7 + 3 × 4)/(2 + 3)​, (2 × 8 + 3 × 13)/(2 + 3)​)`

= `((14 + 12)/5, (16 + 39)/5)`

= `(26/5, 55/5)`

= `(26/5, 11)`

Here M = `(26/5, 11)`