If a Right Angled Isosceles Triangle Has an Area of 2x^2 + 2x + ½. GMAT Problem Solving

Question: If a right angled isosceles triangle has an area of 2x^2 + 2x + ½. What is the perimeter?

1. 3(x+1)
2. \(\sqrt{2}\)(x^2+1)
3. (\(\sqrt{2}\)x-1)(2x+1)
4. (2+\(\sqrt{2}\))(2x+1)
5. 2x+\(\sqrt{2}\)

“If a right angled isosceles triangle has an area of 2x^2 + 2x + ½.”- 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 questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

There is only one approach to solve the problem statement.
Given:

• The right angled isosceles triangle has an area of 2x^2 + 2x + ½.

Find out:

• The perimeter

Approach:
We need to consider the three sides of an isleces triangle first. Then, we need to calculate the perimeter. Once the perimeter is calculated, we need to find the area. The area is already given in the question. Hence, we can find the actual area. Once the value of one side is found, we need to put that value in the perimeter to find the actual perimeter.
Let the sides of an isosceles right angled triangle be a, a, \(\sqrt{2}\)a
Hence, the Perimeter
= a + a + \(\sqrt{2}\)a
=(2+\(\sqrt{2}\))a
Now, the area becomes
=>½ * a*a
=>\(a^2\)/ 2
The given area as per the problem statement is
=> 2\(x^2\)+ 2x + ½
Hence, from the above equations, we get:
Area = the given area
\(a^2\)/ 2 = 2\(x^2\)+ 2x + ½
\(a^2\)= 4\(x^2\)+ 4x + 1
\(a^2\) = \((2x+1)^2\)
a = 2x+1
Now, we already found the perimeter at the starting of the solution. The perimeter was (2+2)a.
The required perimeter:
(2+\(\sqrt{2}\))a
Putting the value of a
(2+\(\sqrt{2}\)) (2x+1)

Correct Answer: D

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