Question

In the following figure, calculate the area of the shaded portion:

Solution

In right triangle PSQ,
PQ² = PS² + QS²

| By Pythagoras Theorem
= (12)² + (16)²= 144 + 256 = 400
$rightwards double arrow space space space space PQ equals square root of 400 equals 20 space cm$
New, for $triangle PQR comma$
a = 20 cm
b = 48 cm
c = 52 cm
$therefore space space space space straight s space equals space fraction numerator straight a plus straight b plus straight c over denominator 2 end fraction space space space space space space space space space equals space fraction numerator 20 plus 48 plus 52 over denominator 2 end fraction equals 60 space cm therefore space space Area space of space increment PQR space space space space space space space space space equals square root of straight s left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root space space space space space space space space space equals square root of 60 left parenthesis 60 minus 20 right parenthesis left parenthesis 60 minus 48 right parenthesis left parenthesis 60 minus 52 right parenthesis end root space space space space space space space space space equals space square root of 60 left parenthesis 40 right parenthesis left parenthesis 12 right parenthesis left parenthesis 8 right parenthesis end root space space space space space space space space space equals space square root of left parenthesis 6 cross times 10 right parenthesis left parenthesis 4 cross times 10 right parenthesis left parenthesis 6 cross times 2 right parenthesis left parenthesis 8 right parenthesis end root space space space space space space space space space equals space 6 cross times 10 cross times 8 equals 480 space cm squared Area space of space triangle PSQ equals 1 half straight x space Base space straight x space Altitude space space space space space space space space space space space space space space space space space space space space space equals space 1 half straight x space 16 space straight x space 12 space equals space 96 space cm squared$

∴ Area of the shaded portion
= Area of ΔPQR – Area of ΔPSQ
= 480 – 96 = 384 cm².

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Statistics

In the following figure, calculate the area of the shaded portion:

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