Question: ES A certain right triangle has sides of length x, y, and z, where x < y < z. If the area of this triangular region is 1, which of the following indicates all of the possible values of y?

A. y > √2
B. √3/2 < y < √2
C. √2/3 < y < √3/2
D. √3/4 < y < √2/3
E. y < 3√4

Answer: A

Solution and Explanation:

Approach Solution 1:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with geometry. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
Area of triangle = base * height * ½
The number of right triangles with an area of one is infinity.
Finding a triangle that satisfies the criteria is one way to proceed, after which we can draw any conclusions.
One such right triangle is as follows:

Aside: As the question does not inquire as to the range of potential values for z, there is no need to compute the actual value of z, which is 5.
This triangle satisfies the conditions that x, y, and z are true.
Y = 2 applies to this triangle.
When we verify the possible answers, we find that just one (response choice A) permits y to equal 2.
Correct option: A

Approach Solution 2:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with geometry. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
If x > y > z, then our RIGHT triangle appears to be as follows (where the hypotenuse is always the longest side)

I see that B, C, D, and E all yield a MAXIMUM value of y when we scan the answer choices (ALWAYS examine the answer choices before completing any computations!).
Given that there is no restriction on the length of each triangle leg, this should come as a bit of a surprise.
GIVEN: The triangle's area is 1; its area is equal to (base)(height)/2.
Hence, the following triangle with area 1 is conceivable:

As an aside, area is equal to (10)(0.2)/2 = 2/2 = 1 (voila!
This triangle's x, y, and z values are 0.2, 10, and an amount more than 10.
We can rule out response options B, C, D, and E as y may equal 10.
Correct option: A

Approach Solution 3:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with geometry. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
Assuming that this triangle's area is 1, x, y, and z.
Hence, the triangle's area is equal to half of x and y, where x and y can either be the base or the height because the triangle is right-angled and z must be the hypotenuse.
Let's now determine y's minimum value (or maximum value of x).
Given that it is a right-angled triangle, the greatest value of x will occur in the triangle with the dimensions 45:45:90.
=> x=y As a result, x*y=2 = x*x=2. (i.e., the minimum value of y) = x2=2 = x=2
Yet since y>x is a known fact, y>2 follows.
Correct option: A

“A certain right triangle has sides of length x, y, and z, where x < y" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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