Quadrilateral PQRS has PQ = QR = RS = 5, If angle R = 90° and angle Q

Question

Quadrilateral PQRS has PQ = QR = RS = 5, If angle R = 90° and angle Q = 135° what is the area of the quadrilateral?

A.  \frac{25 + \sqrt{2}}{2}

B.  \frac{1 + 25\sqrt{2}}{2}

C.  \frac{25(1 + \sqrt{2})}{2}

D.  25 + \sqrt{2}

E.  25 + 25\sqrt{2}

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chetan2u · Answer

Let us draw the quadrilateral -

Area of the quad PQRS = Area of  	riangle QRS  + Area of  	riangle QPS

Area of  	riangle QRS  =  \frac{1}{2}*5*5=\frac{25}{2}

riangle QPS ......
 \angle PQS=135-45=90 , so it is a right angled triangle, and RS =  \sqrt{5^2+5^2}=5\sqrt{2} 
Area of  	riangle QPS  =  \frac{1}{2}*5*5\sqrt{2}=\frac{25\sqrt{2}}{2}

Area of the quad PQRS = Area of  	riangle QRS  + Area of  	riangle QPS = \frac{25}{2}+\frac{25\sqrt{2}}{2} = \frac{25(1 + \sqrt{2})}{2}

C

CEdward · Answer

This question was basically impossible if you had no idea what the shape of that thing looked like. Does GMAT throw these kinds of 'imagination' questions?

veejoy · Answer

All, please check my work but here is how I solved...

Draw line parallel to RS from Q to opposite side, calling the new point T. New angle T is 90 because R is 90 and therefore S is 90. Supplementary angle above T is also 90.

This splits parallelogram into rectangle and a triangle above it.

Since original parallelogram has R=90, S=90 and Q=135, P must be 45.

If P is 45 and T's supplementary angle is 90, angle above Q is also 45. This creates a 45-45-90 triangle, TPQ. Legs are TP and QT. Hypotenuse is PQ.

Since RS=5, then QT=5 (since TS perpendicular to RS). Since QT opposite P, which is 45, TP is also 5 since its opposite a 45 degree angle as well.

Applying Pythagorean Theorem, 5^2 + 5^2 = PQ^2. PQ is 5 sq root 2. Therefore QR is same length since PQ=QR.

Therefore, to get area of rectangle we multiply RS (5) * QR (5 sq root 2) for 25 sq root 2.

Add that to area of triangle (5*5)/2

(25 sqroot 2 +25)/2

This factors to  /2

Option C.

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TestPrepUnlimited · Answer

To draw this quadrilateral, first, we are given angles R and Q. If we focus on point R, R is connected by two lengths of 5 and they create an angle of 90 degrees. Similarly, Q is connected by two lengths of 5 and they create an angle of 135 degrees. Then we connect PS and we have our graph.

Notice that triangle QRS is a right triangle, if we connect QS it splits angle PQR into a 90 and 45-degree angle, so triangle PQS is also a right triangle.

QS = 5\sqrt{2} , the areas are  \frac{5^2}{2}  and  \frac{25\sqrt{2}}{2} .

Ans: C

ThatDudeKnows · Answer

The explanations already in the thread are a-okay. Just want to point out that once you draw the figure, you can Ballpark the actual area. The bottom part is almost a full 5x5 square. That's 25. Then we have to account for the top part, which looks like maybe about a quarter the size of the square. Okay, 25+6 = 31.

A.  \frac{25 + 1.4}{2} = \frac{26}{2} = 13 
Note even close.

B.  \frac{1 + 25(1.4)}{2}=\frac{1 + 35}{2}=\frac{36}{2}=18 
Not even close.

C.  \frac{25(1 + 1.4)}{2}=\frac{25(2.4)}{2} = \frac{60}{2} = 30 
Keep it.

D.  25 + 1.4 = 26 
Close but clearly too small.

E.  25 + 25(1.4) = 25+35 = 60 
Not even close.

Answer choice C.

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Quadrilateral PQRS has PQ = QR = RS = 5, If angle R = 90° and angle Q

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Quadrilateral PQRS has PQ = QR = RS = 5, If angle R = 90° and angle Q

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