Question:

A quadrilateral ABCDABCD is inscribed in a circle such that AB:CDAB :CD = 2:12:1 and BC:AD=5:4BC:AD = 5: 4. If ACAC and BDBD intersect at the point EE,then AE:CEAE:CE equals

Updated On: Aug 16, 2024
  • 2 :1
  • 5: 8
  • 8 : 5
  • 1: 2
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The Correct Option is C

Solution and Explanation

A quadrilateral ABCD is inscribed in a circle such that AB:CD:2:1

Given that ABCDABCD is a cyclic quadrilateral.
ADB=ACB ∠ADB = ∠ACB            (Angle subtended by chord on the same side of arc)
DAC=DBC ∠DAC = ∠DBC            (Angle subtended by chord on the same side of arc
AED⇒ △AED  and BEC△BEC are similar triangles.
Similarly, AEB△AEB and DEC△DEC are also similar using AA similarity property.
Given that,
AB:CD=2:1AB : CD = 2:1
and BC:AD=5:4BC: AD = 5:4
AEBE=ADBC=45\frac {AE}{BE} = \frac {AD}{BC} = \frac 45          (Similar Triangles AED△AED and BEC△BEC)

BECE=ABCD=21\frac {BE}{CE} = \frac {AB}{CD} =\frac 21         (Similar Triangles AEB△AEB and DEC△DEC)
On multiplying both,
AECE=85\frac {AE}{CE} = \frac 85

So, the correct option is (C): 8:58:5

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