In the figure shown, triangle PQR is inscribed in the circle. What is

Solution

Step 1- Given and To find

Given

• Triangle PQR is inscribed in the circle

To find

• The radius of the circle

Let us move on to analyze the statements

Step 2: Analysing individual statements

Statement 1: The perimeter of the triangle PQR is 60

• This statement does not give any information about the radius.

Hence, we can not find the answer from this statement.

Statement 2: The ratio of the lengths of QR, PR, and PQ respectively, is 3 : 4 : 5.

• So, QR = 3x, PR = 4x , and PQ = 5x

o And, 3x^2 + 4x^2 = 5x^2
o So, PQR is a right-angled triangle.
• So, PQ is diameter of the circle.
• Hence radius = 5x/2

But, we do not know the value of x
Hence, we can not find the answer from this statement.

We can not find answer using individual statements. Let’s combine both the statements.

Step 3: Combining both the statements

From statement 1: Perimeter of triangle PQR = 60
From statement 2: Radius = 5x /2

Using both 3x + 4x +5x = 60
From this equation, we can find x and then radius can be found.


Hence, option C is the correct answer.
Correct Answer: Option C

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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Since we have 3 variables (PQ, QR and RP) and 0 equations, E is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since QR:PR:PQ=3:4:5, the triangle PQR is a right triangle and we have QR=3k, PR=4k, PQ=5k from condition 2)
The perimeter is 3k+4k+5k=12k=60 or k = 5.
Since a triangle PQR is a right triangle, PQ is a diameter of the circle.
PQ = 25 and the radius is 25/2.

Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.

Hi,
I understood your method, however, I used a different way and arrived at a different answer. Could you please tell me why this is incorrect.
Concluding from stmt 2, pqr is a rt triangle and since PQ ( hypotenuse)is the diameter of the circle = 2r. Now in a 30-60-90 triangle, hypotenuse is the side which is in the ratio of x:xrt3:2x, can we not conclude that PR and QR will be in that ratio.
Now combining stmt 1&2, r+rrt3+2r=60. r = 13 (approx). We can get a value of r from this. Is my inference of a 30-60-90 triangle wrong?

Thanks