Welcome to Future Study Point! In this post, we provide a complete step-by-step solution to Class 9 Maths Chapter 1 orienting-yourself Q11 solution from the End-of-Chapter Exercises of Class 9 Mathematics – Chapter 1: Orienting Yourself: Uses of Coordinates.
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Question Statement
Let P, Q be points of trisection of AB, with P closer to A, and Q closer to B. Using your knowledge of how to find the coordinates of the midpoint of a segment, how would you find the coordinates of P and Q? Do this for the case when the points are A (4, 7) and B (16, –2).
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Concept & Strategy
Points of trisection divide a line segment AB into three equal parts. That is:
AP = PQ = QB
The question specifically asks us to use the midpoint concept rather than direct section formulas. We can observe two key geometric midpoint relationships:
- Since AP = PQ, P is the midpoint of the segment AQ.
- Since PQ = QB, Q is the midpoint of the segment PB.
Step-by-Step Solution
1. Given Coordinates & Setup
- Endpoint A = (4, 7)
- Endpoint B = (16, −2)
- Let point P = (x1, y1)
- Let point Q = (x2, y2)
Concept: Points P and Q divide AB into three equal parts (AP = PQ = QB). Thus, P is the midpoint of AQ and Q is the midpoint of PB.
2. Finding the x-coordinates (x1 and x2)
Using the midpoint formula for the x-coordinates:
For P as the midpoint of AQ:
x1 = (4 + x2) / 2
2x1 = 4 + x2
x2 = 2x1 − 4 — (Equation 1)
For Q as the midpoint of PB:
x2 = (x1 + 16) / 2
2x2 = x1 + 16 — (Equation 2)
Substitute Equation 1 into Equation 2:
4x1 − 8 = x1 + 16
3x1 = 24
x1 = 8
Substitute x1 = 8 back into Equation 1:
x2 = 12
3. Finding the y-coordinates (y1 and y2)
Using the midpoint formula for the y-coordinates:
For P as the midpoint of AQ:
y1 = (7 + y2) / 2
2y1 = 7 + y2
y2 = 2y1 − 7 — (Equation 3)
For Q as the midpoint of PB:
y2 = (y1 + (−2)) / 2
2y2 = y1 − 2 — (Equation 4)
Substitute Equation 3 into Equation 4:
4y1 − 14 = y1 − 2
3y1 = 12
y1 = 4
Substitute y1 = 4 back into Equation 3:
y2 = 1
Final Coordinates
Point P = (8, 4) | Point Q = (12, 1)
Watch the full Solution of Class 9 Maths Ganita Manjari chapter 1 Orienting Yourself: Uses of Coordinates
Click the link–Class 9 Maths Chapter 1 Orienting Yourself: Uses of Coordinates
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