Class 9 Maths Chapter 7 Triangles – Exercise 7.2 NCERT Solutions
Introduction
Exercise 7.2 explores triangle inequalities. You’ll learn how to prove that the sum of any two sides of a triangle is greater than the third side, and how the larger angle lies opposite the longer side.
Key Concepts
Triangle Inequality Theorem:
Angle‑Side Relationship:
Larger side lies opposite larger angle.
Smaller side lies opposite smaller angle.
Solved Questions (Step by Step)
Q1. In ΔABC, AB = 5 cm, BC = 7 cm, AC = 9 cm. Verify triangle inequality.
Solution:
AB + BC = 12 > AC (9).
BC + AC = 16 > AB (5).
AC + AB = 14 > BC (7).
Verified.
Q2. In ΔPQR, PQ = 4 cm, QR = 6 cm, PR = 9 cm. Check if triangle possible.
Solution:
PQ + QR = 10 > PR (9).
QR + PR = 15 > PQ (4).
PR + PQ = 13 > QR (6).
Triangle possible.
Q3. In ΔXYZ, XY = 3 cm, YZ = 4 cm, ZX = 8 cm. Check validity.
Solution:
XY + YZ = 7 < ZX (8).
Condition fails.
Triangle not possible.
Q4. In ΔABC, AB = 7 cm, BC = 10 cm, AC = 12 cm. Verify inequality.
Solution:
AB + BC = 17 > AC (12).
BC + AC = 22 > AB (7).
AC + AB = 19 > BC (10).
Verified.
Q5. In ΔPQR, PQ = 5 cm, QR = 5 cm, PR = 5 cm. Verify inequality.
Solution:
Equilateral triangle.
Any two sides sum > third side.
Verified.
Q6. In ΔLMN, LM = 6 cm, MN = 8 cm, LN = 13 cm. Check validity.
Solution:
LM + MN = 14 > LN (13).
MN + LN = 21 > LM (6).
LN + LM = 19 > MN (8).
Triangle possible.
Q7. In ΔABC, AB = 2 cm, BC = 3 cm, AC = 6 cm. Check validity.
Solution:
AB + BC = 5 < AC (6).
Condition fails.
Triangle not possible.
Q8. In ΔXYZ, XY = 9 cm, YZ = 10 cm, ZX = 5 cm. Verify inequality.
Solution:
XY + YZ = 19 > ZX (5).
YZ + ZX = 15 > XY (9).
ZX + XY = 14 > YZ (10).
Verified.
Q9. In ΔPQR, PQ = 12 cm, QR = 15 cm, PR = 20 cm. Verify inequality.
Solution:
PQ + QR = 27 > PR (20).
QR + PR = 35 > PQ (12).
PR + PQ = 32 > QR (15).
Verified.
Q10. In ΔABC, AB = 8 cm, BC = 6 cm, AC = 14 cm. Check validity.
Solution:
AB + BC = 14 = AC.
Condition requires >, not =.
Triangle not possible.
Q11. In ΔXYZ, XY = 7 cm, YZ = 24 cm, ZX = 25 cm. Verify inequality.
Solution:
XY + YZ = 31 > ZX (25).
YZ + ZX = 49 > XY (7).
ZX + XY = 32 > YZ (24).
Verified.
Q12. In ΔPQR, PQ = 9 cm, QR = 40 cm, PR = 50 cm. Check validity.
Solution:
PQ + QR = 49 < PR (50).
Condition fails.
Triangle not possible.
Q13. In ΔABC, AB = 15 cm, BC = 20 cm, AC = 25 cm. Verify inequality.
Solution:
AB + BC = 35 > AC (25).
BC + AC = 45 > AB (15).
AC + AB = 40 > BC (20).
Verified.
Q14. In ΔLMN, LM = 10 cm, MN = 12 cm, LN = 22 cm. Check validity.
Solution:
LM + MN = 22 = LN.
Condition fails.
Triangle not possible.
Q15. In ΔXYZ, XY = 11 cm, YZ = 13 cm, ZX = 15 cm. Verify inequality.
Solution:
XY + YZ = 24 > ZX (15).
YZ + ZX = 28 > XY (11).
ZX + XY = 26 > YZ (13).
Verified.
FAQs (10 for Exercise 7.2)
Q: What is the triangle inequality theorem? A: The sum of any two sides of a triangle is greater than the third side.
Q: Can equality hold in triangle inequality? A: No, it must be strictly greater.
Q: What happens if inequality fails? A: The triangle cannot be formed.
Q: What is the relation between sides and angles? A: Larger side lies opposite larger angle.
Q: Can a triangle have sides 2, 3, 6? A: No, because 2 + 3 < 6.
Q: Can a triangle have sides 5, 5, 5? A: Yes, equilateral triangle satisfies inequality.
Q: Why is triangle inequality important? A: It ensures triangle formation and helps in proofs.
Q: Can triangle inequality be extended to polygons? A: Yes, but triangles are the basic case.
Q: What is the exterior angle property relation to inequality? A: Exterior angle > each interior opposite angle.
Q: How does inequality help in construction? A: It ensures chosen side lengths can form a triangle.
Conclusion
Exercise 7.2 has 15 questions that strengthen your understanding of triangle inequalities and side‑angle relationships. This builds the foundation for proving geometric theorems and solving advanced problems in triangles.
Visit:www.fuzymathacademy.com
