Question

A square with side lengths 5 centimetres is positioned with one apex at the centre of a second square (which has side lengths of 4 centimetres). The longest side of the overlapping (shaded) quadrilateral region has a length of 3 centimetres. Calculate the area of the shaded region. Show your full working-out together with an appropriately labelled sketch.

 

 

Answer 1

Draw OC ⊥ CD and OA ⊥ AB

Let

∠ COD = θ

∴ ∠ DOA = 90 – θ                                    (∵ ∠ COD + ∠ DOA = 90°)

∵∠ DOB = 90°

∴ ∠ DOA + ∠ AOB = 90°

90° - θ + ∠ AOB = 90°

∠ AOB = 90°

Now, In △COD and △AOB

∠ C = ∠ A = 90°

∠ COD = ∠ AOB = θ

OC = OA                                                      (Half side of Square)

∴ By AAS

△COD ≅  △AOB

∴ ar (COD) = ar (AOB)

Now,

ar (DEBO) = ar(DEAO) + ar(AOB)

= ar(DEAO) + ar(COD)

= ar(CEAO) = CE × AE

= 2 × 2

= 4 cm 2

∴ Area of shaded region is 4 cm 2 .

Area of shaded region is 4 cm 2 .