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ABCD is a rectangle of dimensions 6 cm x 8 cm. DE and BF are the perpendicular bisectors drawn on the diagonal of the rectangle. What is the ratio of the area of the shaded to that of the unshaded region?

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Ramesh answered 3 year(s) ago

What is the ratio of the area of the shaded portion to the area of the unshaded portion?

ABCD is a rectangle of dimensions 6 cm x 8 cm. DE and BF are the perpendicular bisectors drawn on the diagonal of the rectangle. What is the ratio of the area of the shaded to that of the unshaded region?



Class-X Maths

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Asked by Shivika

Nov 20

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Ramesh , SubjectMatterExpert

Member since Apr 01 2014

Sol:
Given that DE and BF are perpendiculars drawn on the diagonal AC.
⇒ ∠AED = ∠CED = ∠BFA = ∠BFC = 90°

Each angle of a rectangle is a right angle.
∴ ∠DAB = 90°
∠DAE + ∠EAB = 90°
∠EAB = (90° - ∠DAE)

Consider ΔAFB and ΔBFC,
∠BFA = ∠BFC = 90° [Given]

∠EAB = (90° - ∠DAE)
⇒ ∠BCF = ∠DAE     [Complementary angle of ∠EAB]
⇒ ∠CBF = (90° - ∠DAE) [Complementary angle of ∠BCF]
∴ ∠EAB = ∠CBF

Therefore, ΔAFB and ΔBFC are similar.

Area of shaded region = 2(Area of ΔAFB)
Area of unshaded region = 2(Area of ΔBFC)

Ratio of areas of shaded and unshaded regions
= 2(Area of ΔAFB) : 2(Area of ΔBFC)
= (Area of ΔAFB) : (Area of ΔBFC)

Ratio of areas of similar triangles is equal to the ratio of the squares of the corresponding sides.

= (8)2 : (6)2
= 64 : 36
= 16 : 9

Answer 2

Answer:

16 : 9

Step-by-step explanation: