Question 25A: In an equilateral ∆ABC, D is a point on side BC such BD = \\frac{1}{3})BC. Prove that 9(AD)^{2} = 7(AB)^{2}

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ABC is an equilateral triangle.

BD = \\frac{1}{3}) BC

Let the side of equilateral triangle measure 'a' units.

Draw AH perpendicular to BC.

In right triangle AHC, AH^{2} + HC^{2} = AC^{2}

By SAS rule AHB and AHC are congruent,

HC = \\frac{1}{2}) BC = \\frac{1}{2}) a

In right triangle AHB, AH^{2} = AB^{2} - HB^{2}

AH^{2} = a^{2} - (\\frac{1}{2}) a)^{2} = a^{2} - \\frac{1}{4}) a^{2} = \\frac{3}{4}) a^{2}

In right triangle ADH, AD^{2} = AH^{2} + HD^{2}

HD = BH - BD = \\frac{1}{2})a - \\frac{1}{3})a = \\frac{1}{6})a

∴ AD^{2} = \\frac{3}{4})a^{2} + (\\frac{1}{6})a)^{2} = \\frac{3}{4})a^{2} + \\frac{1}{36})a^{2}

= \\frac{27a^2 + a^2}{36}) = \\frac{28}{36})a^{2} = \\frac{7}{9})a^{2}

∴ AD^{2} = \\frac{7}{9})(AB)^{2}

or 9(AD)^{2} = 7(AB)^{2}

Class 10 Maths

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CBSE Class 10 Maths - 2022

- Real Numbers Revision Class
- Polynomials Revision Videos
- Linear Equations Revision Class
- Quadratic Equations Revision Class
- Arithmetic Progressions Revision Video
- Triangles Revision
- Coordinate Geometry Revision
- Trigonometry Revision Class
- Appl of Trigonometry
- Circles Revision Class
- Areas Related to Circles Revision Videos

1. Section A | 2018 CBSE Class 10 Maths 1 Mark **Questions 1 to 6**

2. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 7** | Real Numbers - Irrational Numbers

3. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 8** | Linear Equations in 2 variables

4. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 9** | Sum of Arithmetic Progressions

5. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 10** | Coordinate Geometry Section Formula

6. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 11** | Probability of Rolling 2 dice

7. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 12** | Probability of selecting numbers

8. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 13** | Real Numbers - LCM & HCF

9. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 14** | Polynomial Factorization

10. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 15A** | Coordinate Geometry - Parallelogram

11. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 15B** | Coordinate Geometry - Area of Triangles

12. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 16** | Quadratic Equations - Speed Time Distance

13. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 17** | Similar & Congruent Triangles

14. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 19** | Trigonometric Ratios

15. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 20** | Areas related to circles - sector area

16. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 21** | Surface Areas & Volumes - Cylinder, hemisphere and cone

17. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 22** | Statistics - Median

18. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 23** | Quadratic Equations _ Speed Distance Time

19. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 24** | Arithmetic Progressions

20. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 25A** | Equilateral & Right Triangles and Congruence

21. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 25B** | Pythagoram Theorem Proof

22. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 27** | Trigonometric Ratios & Identities

23. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 28** | Surface Area of Frustum

24. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 29** | Applications of Trigonometry - Heights & Distances

25. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 30** | Statistics - 2 Alternatives

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