Multiple choice

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

  1. 1 : 2

  2. 5 : 8

  3. 8 : 5

  4. 2 : 1
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C Correct answer
Explanation

Triangles ABE and DCE are similar because angles subtended by the same arcs are equal. The ratio of their sides is AB/CD = 2/1. Therefore, AE/DE = BE/CE = 2/1. Using the property of cyclic quadrilaterals and similar triangles, the ratio AE/CE is derived from the side ratios.

AI explanation

In a cyclic quadrilateral ABCD, the opposite angles sum to 180 degrees, so angle ABC equals angle ADC, making triangles ABE and DCE similar by AA similarity. This similarity implies that the ratio AE to CE is equal to AB squared divided by the product of AD and BC. Given the ratios AB to CD equals 2 to 1 and BC to AD equals 5 to 4, substituting these proportional values gives AB squared as 4 units squared and the product of AD and BC as 5 units squared. The ratio AE to CE becomes 4 divided by 5, which is 8 to 5.

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