In the diagram, triangle PQR has a right angle at Q and a perimeter of GMAT Problem Solving

Question: In the diagram, triangle PQR has a right angle at Q and a perimeter of 60. Line segment QS is perpendicular to PR and has the length of 12.PQ > QR. What is the ratio of the area for the triangle PQS to the area of the triangle RQS?

     A. 3/2
     B. 7/4
     C. 15/8
     D. 16/9
     E.  2

Answer: D

Approach Solution (1):Triangles PQR, PSQ, and QSR are all similar because the internal angles of all these triangles are equal. So, if a,b,c are the lengths of sides of PQR, then the sides of PSQ will be ka,kb,kc and those of QSR will be la,lb,lc where k and l are constants.
Perimeter of PQR is 60
So, a+b+c = 60
Sum of perimeters of PQS and QSR = (k+l)(a+b+c) = 60+2*QS = 60+24 = 84
So, (k+l)(60) = 84 => k+l = 1.4 --eqn(1)
Area of triangle PQR = (1/2)a*b
Area of triangle PSR = (1/2)(ka)*(kb) = (1/2)abk²
Area of triangle QSR = (1/2)(la)*(lb) = (1/2)abl²
Area of PQR = area of PSR + area of QSR
So, ab = (k²+l²)ab
So, k²+ l² = 1 -- eqn(2)
From (1) and (2): k = 0.8, l = 0.6 because PQ>QR
So, ratio of areas is k²/l² = (0.8/0.6)² = 16/9
Correct option: D

Approach Solution (2):
Let us assume that PQ = a, QR = b, and PR = c
By Pythagoras Theorem, in triangle PQR, c² = a² + b²
Area of triangle PQR = (1/2) * a * b = (1/2) * c * 12, which implies ab/2 = 12c/2 or ab = 12c
It is given that the perimeter of PQR = 60, which means a + b + c = 60
PQ > QR implies a > b
We know ab in terms of c, a + b = 60 - c and we also know the value of a² + b² in terms of c². So let us apply the formula of (a + b)² = a² + b² + 2ab so that we get an equation in terms of variable c.
After substituting the values in terms of c, we get,
(60 - c)² = c² + 2 * 12c
3600 + c² - 120c = c² + 24c
3600 = 144c
c = 25
ab = 12c implies ab = 12 * 25 = 300
a + b = 60 - c implies a + b = 60 - 25 = 35 implies b = 35 - a
Solving above 2 equations we get, a(35 - a) = 300
a² - 35a + 300 = 0
a² - 15a - 20a + 300 = 0
a(a - 15) - 20(a - 15) = 0
a = 15, 20 implies b = 20, 15
It is given that a > b, so the only possible value of a = 20 and b = 15.
If an altitude is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangles which are both similar to the original and therefore similar to each other.So, the areas of these triangles will be in the ratio of the square of respective sides.
Since, a = 20, b = 15, so a : b = 20 : 25 = 4 : 3
To have the ratio of areas, we should square the above ratio, so (4 : 3)² = 16 : 9
Correct option: D

Approach Solution (3):
Since we have three different triangles, and two of them have a side length of 12, AND we know the perimeter is a nice round number like 60, it's not a huge stretch to assume that they might all be 3-4-5 triangles.
If we make this assumption, we would conclude that for the biggest one, 3x + 4x + 5x = 60, or 12x = 60. That means x=5. That means the triangle is a 15-20-25 triangle.
This means the hypotenuses (sp?) of the other two 3-4-5 triangles are 15 and 20. This means they are 9-12-15 and12-16-20 triangles respectively. This works well, since we already know that they must share a side of12 (QS).
So, the ratio of the areas is [spoiler](12*16)/(12*9) = 16/9.
Correct option: D

“In the diagram, triangle PQR has a right angle at Q and a perimeter of 60. Line segment QS is perpendicular to PR and has the length of 12.PQ > QR. What is the ratio of the area for the triangle PQS to the area of the triangle RQS?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. 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.

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