Geometry-Concept Problems-1 (GMAT-Problem Solving)

Triangle Inequality Theorem:

Sum of two sides of a triangle must be greater than the measure of the third side

a + b > c                b + c > a                a + c > b

Q1) Given Integers as the lengths of sides of a triangle. Find the maximum and minimum perimeter of the triangle where two of the sides measure 9 and 5?

Solution:

To find maximum perimeter

Sum of the lengths of given sides = 9+5 = 14

Hence maximum length of the third side = 13

(This can be understood from Triangle inequality theorem)

Maximum perimeter = 9+5+13 =27

To find minimum perimeter

For finding minimum perimeter, we can consider 9 as the length of longest side.

As the given sides are 5 and 9, let us consider the third side between 1 and 5.

For meeting the triangle inequality rule, we cannot consider 1 to 4.

Hence the minimum value to be considered for the third side = 5, in which case the three sides    

measure 5, 5 and 9

(Test triangle inequality rule:      5+5>9         5+9>5    9+5>5)

Minimum perimeter = 5+5+9 = 19

The perimeter of the given triangle ranges between 19 and 27

Q2) Which of the following sets of sides won’t form a triangle?

(A)3,3,4                 (B)3,4,5                 (C)3,4,6                 (D)3,4,7                 (E)2,3,4

Solution:

This is an application of Triangle Inequality rule.  Check this rule for the given options one by one. For option D, 3+4 = 7, this is equal to the third side but not more than that. It's violation of the rule. Hence Answer option is (D)