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Rajan@2021
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In the figure given below, ∠ABC=∠DAC and AB=8cm,AC=4cm,AD=5cm. (i) Prove that △ACD is similar to △BCA (ii) Find BC and CD (iii) Find the area of △ACD : area of △ABC.

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This is the basic and conceptual question from similarity chapter in which we are to

(i) Prove that ACD is similar to BCA

(ii) Find BC and CD

(iii) Find the area of ACD : area of ABC.

We have given a figure of triangle in which ∠ABC=∠DAC and AB=8cm,AC=4cm,AD=5cm

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1 Answer

  1. From the question it is given that,
    ABC=DAC
    AB=8cm,AC=4cm,AD=5cm
    (i) Now, consider ACD and BCA
    C=C … [common angle for both triangles]
    ABC=CAD … [from the question]
    So, ACDBCA … [by AA axiom]
    (ii) AC/BC=CD/CA=AD/AB
    Consider AC/BC=AD/AB
    4/BC=5/8
    BC=(4×8)/5
    BC=32/5
    BC=6.4cm
    Then, consider CD/CA=AD/AB
    CD/4=5/8
    CD=(4×5)/8
    CD=20/8
    CD=2.5cm
    (iii) from (i) we proved that, ACDBCA
    area of ACB/area of BCA=AC2/AB2
    =42/82
    =16/64
    By dividing both numerator and denominator by 16, we get,
    =41
    Therefore, the area of ACD : area of ABC is 1:4.

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