# CAT Practice : Coordinate Geometry

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Apply cosine rule!

## Side of a triangle

Q.33: The two sides of a triangle are 8 cm and 9 cm and one angle is 60. Which of the following can be the length of its third side?
I. ${\surd 23}$ cm
II. ${\surd 73}$ cm
III. (4.5 - ${\surd 3.25}$) cm
IV. (4 + ${\surd 33}$) cm
V. (9 + ${\surd 13}$) cm

1. Only II and IV
2. Only I and III
3. Only I, II and V
4. Only II, III and IV

Choice (D). Only II, III and IV

## Detailed Solution

Case I:
Let x be the side opposite the angle

By cosine rule, x2 = 82 + 92 – 2*8*9*cos 60
x2 = 145 – 72 = 73
or x = ${\surd 73}$

Case II:
Let 9 cm be the side opposite the angle

By cosine rule, 92 = 82 + x2 – 2*8*x*cos 60
17 = x2 - 8x
x2 - 8x - 17 = 0
or x = (8 +- ${\surd (64 + 68)}$) / 2
= 4 +- ${\surd 33}$
Only 4 + ${\surd 33}$

Case III:
Let 8 cm be the side opposite the angle

By cosine rule, 82 = 92 + x2 – 2*9*x*cos 60
-17 = x2 - 9x
x2 - 9x + 17 = 0
or x = (9 +- ${\surd (81 - 68)}$) / 2
= 4.5 +- (${\surd 13}$)/2
=4.5 +- ${\surd 3.25}$

Correct Answer: (D). Only II, III and IV

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