Question

# In the figure, ABC; is triangle in which C = 90º and AB = 5 cm. D is a point on AB such that AD = 3 cm and = 60º. Then the length of AC is –

- 5cm
- cm
- cm
- none of these

Hint:

### apply the sine rule to the triangle ACD and the m-n theorem to the triangle ACD.

## The correct answer is: 5cm

### 5cm

Using the sine rule in triangle CDA, we get AC/sin CDA = AD/sin ACD

AC=3 x sin CDA /sin60

Using the m-n property, we can say that

(3+2) cot <CDA = 2 cot 30 – 3 cot 60

cot <CDA = √3/5

sin <CDA = 5/√28

AC = 3x 2 x 5/√3 x √28

AC = 5 √(3/7)

the m-n theorem states that

**(m + n) cot θ = m cot α – n cot ß****this gives us the cotangent of the angle CDA which is further used in the sine rule to find the length of AC**

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