MathematicsAP=3 cm,AR=4.5cm,AQ=6 cm,AB=5 cm andAC=10 cm. Then what is the value of AD and ArΔARQ:ArΔADC?

AP=3 cm,AR=4.5cm,AQ=6 cm,AB=5 cm andAC=10 cm. Then what is the value of AD and ArΔARQ:ArΔADC?


  1. A
    7.5 cm, 9: 25
  2. B
    8 cm, 4: 9
  3. C
    6 cm, 16: 25
  4. D
    5.5 cm, 25: 36 

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    Solution:

    Given, AP=3 cm, AB=5 cm,  AQ=6 cm, AR=4.5cm and  AC=10 cm.
    ⇒ AP AB = AQ AC = 3 5 ............. 1   By the converse of Thales theorem,
    PQ||BC............ 2   In ΔADC  ,
    RQ∥DC [From 2]
    ⇒ AR AD = AQ AC   ⇒ARAD=35 [From 1]
    ⇒AD=53×4.5∵AR=4.5 Therefore, AD=7.5 cm.
    By the property of similar triangles,
      Ar ΔARQ Ar ΔADC = A R 2 A D 2   ⇒ Ar ΔARQ Ar ΔADC = 4.5 7.5 2   ⇒ Ar ΔARQ Ar ΔADC = 3 5 2   ⇒ Ar ΔARQ Ar ΔADC = 9 25  
    Thus, the values of AD = 7.5 cm and ar△ARQ:ar△ADC=9:25.
    Therefore, the correct option is 1.
     
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