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