The Lengths of the Sides of an Obtuse-Angled Triangle are x, y, and z, Where x, y and z are Integers GMAT Problem Solving

Question: The lengths of the sides of an obtuse-angled triangle are x, y, and z, where x, y and z are integers. If xy = 4, what is the value of z?

1. 1
2. 2
3. 3
4. 4
5. 5

‘The lengths of the sides of an obtuse-angled triangle are x, y, and z, where x, y and z are integers.’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “Official Guide for GMAT Reviews”. To solve GMAT Problem Solving examiners measure how well the candidates make analytical and logical approaches to solve numerical problems. In this section, candidates have to evaluate and interpret data from given graphical representation. In this section, mostly one finds out mathematical questions. Five answer choices are given for each GMAT Problem solving question.

Solution and Explanation:

Approach Solution 1:

Let x, y, and z are the three sides of a triangle and are integers. It is mentioned that the triangle has an obtuse angle and x.y = 4
So, Potential (x,y) set = (1,4), (2,2)
If (x,y)=(1,4), then 3 Possible integer z = 4.
However, x^2 + y^2 > z^2 shows that the triangle is not an acute-angled triangle.
If (x,y)=(2,2), then 0 For z=1, x^2 + y^2 > z^2, this does not satisfy the condition of an acute-angled triangle
For z=2, x^2 + y^2 > z^2, this does not satisfy the condition of an acute-angled or equilateral triangle.
For z=3, x^2 + y^2 < z^2, this satisfies the condition of an obtuse-angled triangle.
So, if the triangle has an obtuse angle, then x=2, y=2, and z=3
Therefore option C is the correct answer.

Correct Answer: C

Approach Solution 2:

Case 1: It is given in the question that 2 known sides and 1 unknown side. Let the known sides be A, B, and the unknown side is X. So, according to the triangle inequality theorem it can be said that
(A - B) < X < (A + B)

Case 2: It is given that any obtuse triangle, whose one angle is greater than 90 degrees and the other two angles are less than 90 degrees - given that L
= length of longest side across from the obtuse angle and P and Q are the lengths of the 2 shorter sides.
So the following inequality must hold true for an obtuse triangle:
(L)^2 > (P)^2 + (Q)^2
Given that X, Y, and Z are integer side lengths and X * Y = 4, we have two possible scenarios:

Scenario 1: X = 1 and Y = 4 and Z is the unknown side
Therefore the Triangle inequality theorem follows:
(4 - 1) < Z < (4 + 1)
3 < Z < 5
The only integer value for Z must be 4
Z = 4; X = 1; Y = 4
Inequality for Obtuse Triangle:
(4)^2 > (1)^2 + (4)^2 this is not true. Thus, the sides of X and Y are not 1 and 4

Scenario 2. X = 2 and Y = 2 and Z is unknown
(2 - 2) < Z < (2 + 2)
0 < Z < 4
Z can equal 1, 2, or 3
In order to be an obtuse triangle, the only possible value for Z is 3 because then Z would be the longest side across from the obtuse angle and:
(3)^2 > (2)^2 + (2)^2
9 > 8 This satisfies the condition
If Z = 1 or 2, then we can not satisfy this inequality for an obtuse triangle
Therefore Z = 3
Therefore option C is the correct answer.

Correct Answer: C

Approach Solution 3

it is given in the problem that x,y &z are integers. So, as we have xy =4, we can say the possible values be 2 x 2 or 4 x 1.
Case 1: 4 x 1
Here we get the third side can only be 4
But this doesn't satisfy the condition of a^2 + b^2 < c^2

Case 2: 2 x 2
Third side in this case can take three values = 1,2,3
If we consider 2 then it will become equilateral
If we consider 1 it is not possible either. So, 3 as it satisfies the above equation as well
So option C is correct.

Correct Answer: C

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