QUESTION 10
Easy
Given a square ABCD, let P be a point within it. Join PA, PB, PC, PD (figure). What is the ration of the areas of the red region (ΔPAB and ΔPD) and the green region (ΔPBC and ΔPDA)?
SOLUTION
Area of square\( = ΔPAB\, +\, ΔPBC\, + \,ΔPCD\, +\, ΔPDA\)
Since \(AB || CD\),
the sum \(ΔPAB \,+ \,ΔPCD\) occupies half the square.
Similarly,
\(ΔPBC \,+ \,ΔPDA\) occupies the remaining half.
Therefore,
\(Area(ΔPAB\,+\,ΔPCD\)\,=\,Area(ΔPBC\,+\,ΔPDA)\)
The ratio is 1 : 1.
Since \(AB || CD\),
the sum \(ΔPAB \,+ \,ΔPCD\) occupies half the square.
Similarly,
\(ΔPBC \,+ \,ΔPDA\) occupies the remaining half.
Therefore,
\(Area(ΔPAB\,+\,ΔPCD\)\,=\,Area(ΔPBC\,+\,ΔPDA)\)
The ratio is 1 : 1.
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Final Answer : The ratio is 1 : 1.
Concept Note
A square is a parallelogram.
In any parallelogram, the sum of the areas of triangles on one pair of opposite sides equals the sum of the areas on the other pair.