CAT 2023 Slot 1 QA Question & Solution
GeometryHard
Question
A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : 1 and BC : AD = 5 : 4. If AC and BD intersect at the point E, then AE : CE equals
Options
2 : 1
1 : 2
8 : 5
5 : 8
Solution
Given ABCD is a cyclic quadrilateral.
Angle ADB = Angle ACB (Angle subtended by chord on the same side of arc)
Angle DAC = Angle DBC (Angle subtended by chord on the same side of arc)
=> Triangles AED and BEC are similar triangles
Similarly triangles AEB and DEC are also similar using AA similarity property.
Now, given that AB : CD = 2 : 1 and BC : AD = 5 : 4
AE/BE = AD/BC = 4/5 (Similar Triangles AED and BEC)
BE/CE = AB/CD = 2/1 (Similar Triangles AEB and DEC)
Multiplying both, we get AE/CE = 8/5.