Geometry - CAT 2011 Sample Questions

Question 1

In a triangle ABC, the lengths of the sides AB and AC equal 17.5 cm and 9 cm respectively. Let D be a point on the line segment BC such that AD is perpendicular to BC. If AD = 3 cm, then what is the radius (in cm) of the circle circumscribing the triangle ABC?

(1) 17.05
(2) 27.85
(3) 22.45
(4) 32.25
(5) 26.25

Correct Answer is 26.25 - Choice (5)

Explanation

Area of a triangle = , where a, b and c are lengths of the sides of the triangle and R is the radius of the circle circumscribing the triangle

In this example, area of the triangle = 

The area of any triangle can also be expressed as , as the altitude AD to the side BC is 3 cms.

Equating the two and solving for R, we get R = 26.25 cms

Question 2

How many scalene triangles exist whose sides a, b and c are natural numbers less than 8?

(A) 13
(B) 14
(C) 15
(D) 16

Correct Answer is 13 - Choice (A). 

Explanation

This is a classic trial and error question. To start with let us assume a < b < c

a cannot be 1 because a + b will not be greater than c for integer values of a, b and c. The basic property for a set of 3 lines to form a triangle is that the sum of any two sides should be greater than the third side.

The smallest value a can take is 2

Triangles when a = 2 are

1. 2, 3, 4
2. 2, 4, 5
3. 2, 5, 6
4. 2, 6, 7

Triangles when a = 3 are

5. 3, 4, 5
6. 3, 4, 6
7. 3, 5, 6
8. 3, 5, 7
9. 3, 6, 7

Triangles when a = 4 are

10. 4, 5, 6
11. 4, 5, 7
12. 4, 6, 7

There is only one triangle possible when a = 5, i.e., 5, 6, 7

Totally there are 13 triangles possible.

One needs to be very careful in these types of brute-force enumeration questions. If under time-pressure, it is probably best to skip such questions.