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Coordinate Geometry
CH07 Question Bank
CBSE_CH07_QUESTION_BANK
Grade 10
Question:
[Case Study]
A football coach is marking player positions on a coordinate grid representing the field (in metres). The left winger starts at $A(2,5)$ and the coach wants them to run in a straight line to support the striker at $B(11,17)$.
(a) Find the total distance the winger must run from A to B. [1 Mark] (b) The coach wants the winger to pause at a point P dividing AB in the ratio 2:1 (from A) to receive a pass. Find the coordinates of P. [1 Mark] (c) Find the distance AP that the winger runs before pausing. [1 Mark] (d) A defender stationed at $D(2,17)$ wants to know if they are closer to A or to B. Determine which, with working. [1 Mark]
Step-by-Step Solution
Key Concept: Case study on coordinate geometry.
(a) Find the total distance the winger must run from A to B. [1 Mark] $AB=\sqrt{(11-2)^2+(17-5)^2}=\sqrt{81+144}=\sqrt{225}=15$ m. [1.0 Mark]
(b) The coach wants the winger to pause at a point P dividing AB in the ratio 2:1 (from A) to receive a pass. Find the coordinates of P. [1 Mark] $P_x=\dfrac{2(11)+1(2)}{2+1}=\dfrac{22+2}{3}=8$; $P_y=\dfrac{2(17)+1(5)}{3}=\dfrac{34+5}{3}=13$. So $P=(8,13)$. [1.0 Mark]
(c) Find the distance AP that the winger runs before pausing. [1 Mark] $AP=\sqrt{(8-2)^2+(13-5)^2}=\sqrt{36+64}=\sqrt{100}=10$ m. [1.0 Mark]
(d) A defender stationed at $D(2,17)$ wants to know if they are closer to A or to B. Determine which, with working. [1 Mark] $DA=\sqrt{(2-2)^2+(17-5)^2}=\sqrt{0+144}=12$ m; $DB=\sqrt{(2-11)^2+(17-17)^2}=\sqrt{81+0}=9$ m. Since $DB
Correct Answer:$AB=\sqrt{(11-2)^2+(17-5)^2}=\sqrt{81+144}=\sqrt{225}=15$ m. [1.0 Mark] | $P_x=\dfrac{2(11)+1(2)}{2+1}=\dfrac{22+2}{3}=8$; $P_y=\dfrac{2(17)+1(5)}{3}=\dfrac{34+5}{3}=13$. So $P=(8,13)$. [1.0 Mark] | $AP=\sqrt{(8-2)^2+(13-5)^2}=\sqrt{36+64}=\sqrt{100}=10$ m. [1.0 Mark] | $DA=\sqrt{(2-2)^2+(17-5)^2}=\sqrt{0+144}=12$ m; $DB=\sqrt{(2-11)^2+(17-17)^2}=\sqrt{81+0}=9$ m. Since $DB<DA$, the defender is closer to B. [1.0 Mark]
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