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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
- a)1 : 2
- b)5 : 8
- c)8 : 5
- d)2 : 1
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : ...
Given Information:
- Quadrilateral ABCD is inscribed in a circle.
- AB : CD = 2 : 1 and BC : AD = 5 : 4.
Approach:
- Use properties of inscribed quadrilaterals.
- Apply the intersecting chords theorem.
Explanation:
Step 1: Apply Intersecting Chords Theorem
- According to the intersecting chords theorem, the product of the lengths of the chords of a circle that intersect inside the circle is equal.
- Therefore, AB * BE = CB * DE and AD * AE = DC * CE.
Step 2: Use Ratios Given
- From the given ratios, we have AB : CD = 2 : 1 and BC : AD = 5 : 4.
- Let the lengths be 2x, x, 5y, and 4y respectively.
- Therefore, the lengths of the chords are AB = 2x, CD = x, BC = 5y, and AD = 4y.
Step 3: Calculate the Lengths of BE and DE
- Using AB * BE = CB * DE, we get 2x * BE = 5y * DE.
- BE / DE = 5y / 2x.
Step 4: Calculate the Lengths of AE and CE
- Using AD * AE = DC * CE, we get 4y * AE = x * CE.
- AE / CE = x / 4y.
- Substituting x = 2y (from AB : CD = 2 : 1), we get AE / CE = 2 / 8 = 1 / 4.
- Simplifying, we get AE : CE = 1 : 4 = 8 : 32 = 1 : 4.
Therefore, the ratio of AE : CE is 1 : 4 or 8 : 32 (which simplifies to 1 : 4). Hence, the correct answer is option C, 8 : 5.
- Quadrilateral ABCD is inscribed in a circle.
- AB : CD = 2 : 1 and BC : AD = 5 : 4.
Approach:
- Use properties of inscribed quadrilaterals.
- Apply the intersecting chords theorem.
Explanation:
Step 1: Apply Intersecting Chords Theorem
- According to the intersecting chords theorem, the product of the lengths of the chords of a circle that intersect inside the circle is equal.
- Therefore, AB * BE = CB * DE and AD * AE = DC * CE.
Step 2: Use Ratios Given
- From the given ratios, we have AB : CD = 2 : 1 and BC : AD = 5 : 4.
- Let the lengths be 2x, x, 5y, and 4y respectively.
- Therefore, the lengths of the chords are AB = 2x, CD = x, BC = 5y, and AD = 4y.
Step 3: Calculate the Lengths of BE and DE
- Using AB * BE = CB * DE, we get 2x * BE = 5y * DE.
- BE / DE = 5y / 2x.
Step 4: Calculate the Lengths of AE and CE
- Using AD * AE = DC * CE, we get 4y * AE = x * CE.
- AE / CE = x / 4y.
- Substituting x = 2y (from AB : CD = 2 : 1), we get AE / CE = 2 / 8 = 1 / 4.
- Simplifying, we get AE : CE = 1 : 4 = 8 : 32 = 1 : 4.
Therefore, the ratio of AE : CE is 1 : 4 or 8 : 32 (which simplifies to 1 : 4). Hence, the correct answer is option C, 8 : 5.
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Community Answer
A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : ...
∠DAC=∠DBC
(Angles subtended by the chord DC on the same side.)
∠ADB=∠ACB
(Angles subtended by the chord AB on the same side.)
∠AED=∠BEC
(Vertically Opposite angles.)
(Angles subtended by the chord AD on the same side.)
∠BAC=∠BDC
(Angles subtended by the chord BC on the same side.)
∠AEB=∠DEC
(Vertically Opposite angles.)
Therefore, AE : EC = 8 : 5
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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 equalsa)1 : 2b)5 : 8c)8 : 5d)2 : 1Correct answer is option 'C'. Can you explain this answer?
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