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Coordinate Geometry
CH07 Question Bank
CBSE_CH07_QUESTION_BANK
Grade 10

Question:

Two friends, Aditi and Bhavya, are planting saplings along a coordinate grid representing their school garden. Aditi's sapling is at point $A(2,3)$ and Bhavya's is at point $B(8,11)$. They decide to place a water sprinkler at a point $P$ on segment $AB$ such that $AP:PB=3:5$.

(i) Find the coordinates of $P$.
(ii) Find the total distance $AB$ between the two saplings.
(iii) Verify, using the distance formula, that $AP:PB$ indeed equals $3:5$.

Step-by-Step Solution

Key Concept: Use the section formula for part (i), the distance formula for part (ii), then compute AP and PB directly using the distance formula to verify the given ratio in part (iii).
(i) $P_x=\dfrac{3(8)+5(2)}{3+5}=\dfrac{24+10}{8}=4.25$; $P_y=\dfrac{3(11)+5(3)}{8}=\dfrac{33+15}{8}=6$. So $P=(4.25,6)$. [1.5 Marks]

(ii) $AB=\sqrt{(8-2)^2+(11-3)^2}=\sqrt{36+64}=\sqrt{100}=10$ units. [1.0 Mark]

(iii) $AP=\sqrt{(4.25-2)^2+(6-3)^2}=\sqrt{5.0625+9}=\sqrt{14.0625}=3.75$; $PB=\sqrt{(8-4.25)^2+(11-6)^2}=\sqrt{14.0625+25}=\sqrt{39.0625}=6.25$. [1.5 Marks]

$AP:PB=3.75:6.25=3:5$ ✓, confirming the given ratio (and note $AP+PB=3.75+6.25=10=AB$, consistent with part (ii)). [1.0 Mark]

Correct Answer:
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