A quadrilateral ABCD is inscribed in a circle such that AB:CD=3:2 and BC:AD=4:3. If diagonals AC and BD intersect at E, find AE:CE.
Answer and solution
Correct answer: C
Solution
Answer: C — 9:8
1. In triangles ABE and CDE, the angles at E are vertically opposite, hence have equal sines.
2. By the sine rule, AE/CE=(AB/CD)×[sin∠ABE/sin∠CDE].
3. Because E lies on BD, these angles are ∠ABD and ∠CDB. In the same circumcircle their sines are AD/(2R) and CB/(2R).
4. Therefore AE/CE=(AB×AD)/(CD×CB)=(3/2)(3/4)=9/8.
5. Answer 9:8, option C. The intersecting-chords product alone does not determine this ratio.
Why this method works:
Use the same circumradius R when converting chord lengths to sines. In triangles ABE and CDE the angles at E are vertical, so the sine-rule expressions have a common factor that cancels. This is why the product of adjacent-side ratios appears.
Independent check and common trap:
AB/CD=3/2 and AD/CB=3/4; the second ratio is the reciprocal of the supplied BC/AD. Their product is 9/8, giving AE: CE=9:8. Using 4/3 instead would invert the relevant side pairing.
Apply: For E=AC∩BD in a cyclic quadrilateral, AE/CE=(AB×AD)/(CB×CD). Substitute the two ratios: (3/2)×(3/4)=9/8.
Check: AB/CD=3/2 and AD/CB=3/4; the second ratio is the reciprocal of the supplied BC/AD. Their product is 9/8, giving AE: CE=9:8. Using 4/3 instead would invert the relevant side pairing.
1. In triangles ABE and CDE, the angles at E are vertically opposite, hence have equal sines.
2. By the sine rule, AE/CE=(AB/CD)×[sin∠ABE/sin∠CDE].
3. Because E lies on BD, these angles are ∠ABD and ∠CDB. In the same circumcircle their sines are AD/(2R) and CB/(2R).
4. Therefore AE/CE=(AB×AD)/(CD×CB)=(3/2)(3/4)=9/8.
5. Answer 9:8, option C. The intersecting-chords product alone does not determine this ratio.
Why this method works:
Use the same circumradius R when converting chord lengths to sines. In triangles ABE and CDE the angles at E are vertical, so the sine-rule expressions have a common factor that cancels. This is why the product of adjacent-side ratios appears.
Independent check and common trap:
AB/CD=3/2 and AD/CB=3/4; the second ratio is the reciprocal of the supplied BC/AD. Their product is 9/8, giving AE: CE=9:8. Using 4/3 instead would invert the relevant side pairing.
G Strategy shortcut
Spot: Circles and Cyclic Quadrilaterals.Apply: For E=AC∩BD in a cyclic quadrilateral, AE/CE=(AB×AD)/(CB×CD). Substitute the two ratios: (3/2)×(3/4)=9/8.
Check: AB/CD=3/2 and AD/CB=3/4; the second ratio is the reciprocal of the supplied BC/AD. Their product is 9/8, giving AE: CE=9:8. Using 4/3 instead would invert the relevant side pairing.
G Strategy: Spot: Circles and Cyclic Quadrilaterals. Apply: For E=AC∩BD in a cyclic quadrilateral, AE/CE=(AB×AD)/(CB×CD). Substitute the two ratios: (3/2)×(3/4)=9/8. Check: AB/CD=3/2 and AD/CB=3/4; the second ratio is the reciprocal of the supplied BC/AD. Their product is 9/8, giving AE: CE=9:8. Using 4/3 instead would invert the relevant side pairing.