In the game of archery, a bow is used to shoot arrows at a target board. The player stands far away from the board

Question

Answer the questions based on the given information: 

In the game of archery, a bow is used to shoot arrows at a target board. The player stands far away from the board and aims the arrow so that it hits the board. 

One such board, which is divided into 4 concentric circular sections, is drawn on a coordinate grid as shown. Each section carries different points as shown in the figure. If an arrow lands on the boundary, the inner section points are awarded.

(i) After shooting two arrows, Rohan scored 25 points. Write one set of coordinates for each arrow that landed on the target. 

(ii) If one player's arrow lands on (2, 2.5), how many points will be awarded to the player? Show your work. 

(iii) One of Rohan’s arrow landed on (1.2, 1.6). He wants his second arrow to land on the line joining the origin and first arrow such that he gets 10 points for it. Find one possible pair of coordinates of the second arrow's landing mark. Show your work. 

OR  

(iii) An arrow landed on the boundary and is worth 20 points. The coordinates of the landing mark were of the form (m, -m). Find all such coordinates. Show your steps.

Answer 1

(i) Writes two pairs of possible coordinates such that Rohan scored 20 and 5 points for them. For examples, (1.5, 0) and (3.5, 0).

(ii) Finds the distance of (2, 2.5) from (0, 0) as: 

√(4 + 6.25) = √10.25 units 

Hence, concludes that 5 points will be awarded. 

(iii) Finds the distance of (1.2, 1.6) from the origin as: 

√{(1.2)² + (1.6)²} = 2 units 

Assume that the second arrow lands on the boundary mark and writes that the ratio in which the first arrow divides the origin and the second arrow's landing mark is the ratio of their radii = 2:1. 

Assumes the coordinates of the second arrow's landing mark as (x, y) and uses section formula to write: 

(2x+0/3, 2y+0/3) = (1.2, 1.6) 

Solves the above equation to find the values of the coordinates of the second arrow's landing mark as (1.8, 2.4).

OR 

Identifies the distance between the origin and the coordinate (m, -m) as 2 units and uses the distance formula to write the equation as: 

m² + (-m)² = 2² 

Simplifies the above equation as 2m² = 4. 

Solves the above equation to get y as √2 and (-√2).