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ABCD is a rectangle and O is point inside the rectangle. Point O joins all four vertices of the rectangle. Which statement(s) is/are wrong? A) Sum of area of △AOB and △DOC is equal to sum of area of △BOC and △AOD B) Sum of area of △AOB and △DOC is half the area of rectangle. C) Sum of area of △BOC and △AOD is half the area of rectangle.

ABCD is a rectangle and O is point inside the rectangle. Point O joins all four vertices of the rectangle. Which statement(s) is/are wrong? A) Sum of area of △AOB and △DOC is equal to sum of area of △BOC and △AOD B) Sum of area of △AOB and △DOC is half the area of rectangle. C) Sum of area of △BOC and △AOD is half the area of rectangle. Correct Answer None of these

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Given,

ABCD is a rectangle.

Formula:

Area of the rectangle = Length × Breadth

Calculation:

Let AB = DC = l cm and

BC = AD = b cm, then

Draw a perpendicular OP  AB and OQ  DC,

Similarly, draw a perpendicular OS  BC and OR  AD,

Area of △AOB = (1/2) × AB × OP

Area of △AOB = (1/2) × l × OP ----(1)

Area of △DOC = (1/2) × CD × OQ

Area of △DOC = (1/2) × l × OQ ----(2)

Add equation (1) and equation (2)

Area of △AOB + Area of △DOC = (1/2) × l × (OP + OQ)

OP + OQ = b

Area of △AOB + Area of △DOC = (1/2) × l × b

Area of △AOB + Area of △DOC = (1/2) × Area of rectangle ----(x)

Statement B is correct.

Similarly,

Area of △AOD = (1/2) × AD × OR

Area of △AOD = (1/2) × b × OR     ----(3)

Area of △BOC = (1/2) × BC × OS

Area of △BOC = (1/2) × b × OS     ----(4)

Add equation (3) and equation (4)

Area of △AOD + Area of △BOC = (1/2) × b × (OR + OS)

OR + OS = l

Area of △AOD + Area of △BOC = (1/2) × l × b

Area of △AOD + Area of △BOC = (1/2) × Area of rectangle ----(y)

Statement C is correct.

Equation (x) and equation (y)

Area of △AOB + Area of △DOC = Area of △AOD + Area of △BOC

Statement A is also correct.

∴ All the statements are correct 

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