CAT Quantitative Aptitude Questions | Geometry Questions for CAT - Triangles

The question is from the topic triangles. Two sides and one angle of a triangle are given we have to find out the possible values of the third side. CAT Geometry questions are heavily tested in CAT exam. Make sure you master Geometry problems. Apply the cosine rule here!

Question 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. √23 cm
II. √73 cm
III.(4.5 - √3.25) cm
IV. (4 + √33) cm
V. (9 + √13) cm

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

Explanatory Answer

Method of solving this CAT Question from Triangles: There are three cases.

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 = √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 +- √(64+68)) / 2
= 4 +- √33
Only 4 + √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 +- √(81−68)) / 2
= 4.5 +- (√13)/2
=4.5 +- √3.25

The question is "Choose the correct answer"

Hence, the answer is Only II, III and IV.

Choice D is the correct answer.