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Find the area of the triangle QRS (Tricky)

Find the area of the triangle QRS
                           
A)    90 cm
B)    80 cm
C)    110 cm
D)    55 cm
E)    45 cm        
         

Area of TriangleSolution

• For calculating the area of a triangle, we need to know the basic parameters viz., base and height, and when we substitute them using Area = ½* Base * Height, we will get the area. 

• The base can be any side of a given triangle; it can be QR, RS and QS. Depending on the side we choose, the height will be the altitude drawn from the angle opposite to the base.

So PQ^2  = QR
^2 + PR^2

• Let us take QR as the base here, because we have a right angled triangle involved in it, and the calculation of QR can be based on Pythagoras theorem. 

• We already have the values of PQ and PR values, so let’s use them to get the value of QR.

132 = QR
^2 + 52
169 = QR
^2+ 25

Subtract 25 on both sides

169-25 = QR
^2
144 = QR
^2
12 = QR
QR = 12 cm

• Therefore, the base of triangle QRS = 12 cm. We now need to find the height of the triangle so that we can find the area of it. Here, the height will be the altitude drawn from the vertex S to QR.

• First we draw a perpendicular line from S to QR. Imagining the point of this line on QR as A, we then can see a square formed with points SAQT, with SQ as diagonal as shown in the figure.

Therefore, as the opposite sides of a square are equal, the length of TQ is equal to the length of SA.


• Calculate the length of TQ using Pythagoras theorem. 

TQ^2 + TS
^2 = QS^2
TQ
^2 + 82 = 172
TQ
^2 + 64 = 289

Subtracting 64 on both sides

TQ
^2 = 289-64
TQ
^2 = 225

Applying squaring on both sides

TQ = 15

So the length of TQ = 15 cm

Therefore SA = 15 cm

Hence the altitude is 15 cm

• Now we have the base and height, So we need to find the area of the triangle and we will reach the target

• As Area = ½* base * Height

Area = ½* QR * SA

Area = ½* 12 cm * 15 cm

Area = 90 cm^2

Therefore the area of triangle QRS = 90 cm
^2

• By using Pythagoras theorem we have found the area of the triangle QRS   

As seen here, we have a mix of topics viz., square and right angled triangle.

So the Answer will be Option A (90 cm).

About the Author

GoGMAT, founded in 2009, is an adaptive GMAT preparation platform developed by the best instructors in the industry (with 740+ GMAT scores and strong teaching experience). GoGMAT team consists of high profile members, successful entrepreneurs, experts in educational field, top-notch business school alumni with successful track records as investment bankers and management consultants. Our GMAT experts, university professors and successful entrepreneurs, analyzed GMAT test algorithms for almost 12 months.  We used their knowledge and skills as the basis to create one of the most efficient solutions in the world for GMAT preparation.

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