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# In the Given Figure, Ab and De Are Perpendicular to Bc. 1) Prove that δAbc ∼ δDec 2) If Ab = 6 Cm; De = 4 Cm and Ac = 15 Cm. Calculate Cd. 3) Find the Ratio of Area of δAbc: Area of δDec - ICSE Class 10 - Mathematics

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#### Question

In the given figure, AB and DE are perpendicular to BC.

1) Prove that ΔABC ∼ ΔDEC

2) If AB = 6 cm; DE = 4 cm and AC = 15 cm. Calculate CD.

3) Find the ratio of area of ΔABC: area of ΔDEC

#### Solution

1) In ΔABC and ΔDEC,

∠ABC = ∠DEC = 90°

∠ACB = ∠DCE (Common)

∴ ΔABC ~ ΔDCE (AA criterion)

2) Since ΔABC ∼ ΔDEC,

"AB"/"DE" = "AC"/"CD"

=> 6/4 = 15/"CD" => 6 xx CD = 60 => CD = 60/6 = 10 cm

3) It is known that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

=> (Ar(ABC))/(Ar(DEC)) = "AB"^2/"DE"^2 = 6^2/4^2 = 36/16 = 9/4

=> ar(ABC) : ar(DEC) = 9 : 4`

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Solution In the Given Figure, Ab and De Are Perpendicular to Bc. 1) Prove that δAbc ∼ δDec 2) If Ab = 6 Cm; De = 4 Cm and Ac = 15 Cm. Calculate Cd. 3) Find the Ratio of Area of δAbc: Area of δDec Concept: Similarity of Triangles.
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