TheCATExam
Sign in

CAT 2008 — QA Question 13

Basics of TrianglesEasy
Passage / Data

Directions for next 2 questions:

The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

Consider obtuse-angled triangles with sides 8 cm, 15 cm and x cm. If x is an integer, then how many such triangles exist?

Answer & solution

  • A

    5

  • B

    21

  • 10

  • D

    15

  • E

    14

Solution

We know that for an obtuse triangle of sides a, b and c (where c is the largest side),

a2 + b2 < c2

We also know that for a triangle, a + b > c

Let the third side be x.

These present us with two limiting cases.

Case 1: Let 8 cm and 15 cm be the shorter sides. The value of the largest side (x) must be greater than

82+152=17 cm

Also, x < 8 + 15 = 23.

The possible integer values of x are 18, 19, 20, 21 and 22 cm.

We cannot consider values from 23 onwards because 8 + 15 = 23 and this violates the second condition.

Case 2: Let 8 and x be the shorter sides and 15 cm is the largest side.

The value of the remaining side (x) must be less than

152-82=12.69 cm

Also, x > 15 - 8 = 7

The possible integer values are 12, 11, 10, 9 and 8 cm.

We cannot consider values less than 8 because 7 + 8 = 15 and this violates the second condition.

Thus, we have 10 possible values for x.

Hence, option (c).

CAT 2008 QA Q13: Consider obtuse-angled triangles with sides 8 cm, 15 cm and x cm. If x is an integer, then how many such trian — Solution | TheCATExam