Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles – Exercise 9.2 NCERT Solutions
Introduction
Exercise 9.2 applies the area properties of parallelograms and triangles to practical problems. You’ll learn how to prove equal areas of triangles and parallelograms using base and height arguments, and solve numerical problems based on these properties.
Key Concepts
Triangles on Same Base and Between Same Parallels: Equal in area.
Parallelogram Diagonal Property: Diagonals divide parallelogram into equal triangles.
Area Relation:
Solved Questions (Step by Step)
Q1. Prove that triangles on same base and between same parallels are equal in area.
Solution:
ΔABC and ΔPBC share base BC and lie between parallels AP ∥ BC.
Hence, area(ΔABC) = area(ΔPBC).
Q2. In parallelogram ABCD, prove ΔABC = ΔCDA.
Solution:
Diagonal AC divides parallelogram into two equal triangles.
Q3. In parallelogram PQRS, prove ΔPQR = ΔPRS.
Solution:
Diagonal PR divides parallelogram into two equal triangles.
Q4. In parallelogram LMNO, prove ΔLMN = ΔONM.
Solution:
Diagonal MN divides parallelogram into two equal triangles.
Q5. In ΔABC, D and E are points on AB and AC such that DE ∥ BC. Prove area(ΔADE) = ½ area(ΔABC).
Solution:
DE ∥ BC ⇒ ΔADE and trapezium DEBC share height.
Hence, area(ΔADE) = ½ area(ΔABC).
Q6. In ΔPQR, X and Y are points on PQ and PR such that XY ∥ QR. Prove area(ΔPXY) = ½ area(ΔPQR).
Solution:
By mid‑point theorem, XY ∥ QR.
Hence, area(ΔPXY) = ½ area(ΔPQR).
Q7. In ΔXYZ, M and N are midpoints of XY and XZ. Prove area(ΔXMN) = ¼ area(ΔXYZ).
Solution:
MN ∥ YZ and MN = ½ YZ.
Hence, ΔXMN is ¼ of ΔXYZ.
Q8. In ΔLMN, A and B are midpoints of LM and LN. Prove area(ΔLAB) = ¼ area(ΔLMN).
Solution:
AB ∥ MN and AB = ½ MN.
Hence, ΔLAB is ¼ of ΔLMN.
Q9. In parallelogram ABCD, prove ΔABD = ΔCDB.
Solution:
Diagonal BD divides parallelogram into two equal triangles.
Q10. In parallelogram PQRS, prove ΔPQS = ΔQRS.
Solution:
Diagonal QS divides parallelogram into two equal triangles.
Q11. In parallelogram LMNO, prove ΔLMO = ΔNMO.
Solution:
Diagonal MO divides parallelogram into two equal triangles.
Q12. Prove that area of ΔABC = area of ΔPBC if A and P are on same line parallel to BC.
Solution:
Both triangles share base BC and lie between same parallels.
Q13. In ΔABC, prove that median divides triangle into two equal areas.
Solution:
Median AD divides ΔABC into ΔABD and ΔADC.
Both have equal area.
Q14. In ΔPQR, prove that median divides triangle into two equal areas.
Solution:
Median PM divides ΔPQR into ΔPMQ and ΔPMR.
Both have equal area.
Q15. In ΔXYZ, prove that median divides triangle into two equal areas.
Solution:
Median XM divides ΔXYZ into ΔXMY and ΔXMZ.
Both have equal area.
FAQs (10 for Exercise 9.2)
Q: What is the area property of parallelogram diagonals? A: Diagonals divide it into two equal triangles.
Q: What is the area property of triangles on same base and parallels? A: They have equal area.
Q: How do you prove triangle area equality? A: By showing equal base and equal height.
Q: What is the relation between triangle and parallelogram area? A: Triangle area = ½ parallelogram area.
Q: What is the relation between mid‑point theorem and area? A: Line joining midpoints divides triangle into smaller equal areas.
Q: Why are these properties important? A: They simplify area calculations and proofs.
Q: Can diagonals divide a parallelogram into four equal triangles? A: Yes, when both diagonals are drawn.
Q: What is the sum of angles in a triangle? A: 180°.
Q: What is the sum of angles in a parallelogram? A: 360°.
Q: How do medians affect triangle area? A: Each median divides triangle into two equal areas.
Conclusion
Exercise 9.2 has 15 questions that strengthen your understanding of area properties of parallelograms and triangles. This builds the foundation for solving advanced area problems in geometry.
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