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A rectangle ABCD is inscribed in a circle with centre O. If AC is the diagonal and ∠BAC = 30°, then the radius of the circle will be equal to:
- a)(√3/2)BC
- b)BC
- c)√3BC
- d)2 BC
Correct answer is option 'B'. Can you explain this answer?
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Given Information:
- Rectangle ABCD is inscribed in a circle with center O.
- AC is the diagonal of the rectangle.
- ∠BAC = 30°.
Solution:
1. Properties of Inscribed Angle:
- In a circle, if a quadrilateral is inscribed, then the opposite angles of the quadrilateral are supplementary.
- So, ∠BAD + ∠BCD = 180°.
2. Diagonals of Rectangle:
- In a rectangle, the diagonals are equal in length and bisect each other.
- So, AC = BD.
3. In a Rectangle:
- In a rectangle, the diagonals are perpendicular bisectors of each other.
- ∠BAC = 30° implies ∠BAD = 60°.
- In triangle ABD, ∠ADB = 90°.
- So, ∠BCD = 90° - ∠BAD = 90° - 60° = 30°.
4. In Triangle ACD:
- In triangle ACD, ∠CAD = 90°.
- So, ∠CAD + ∠ACD = 90°.
- ∠ACD = 90° - ∠CAD = 90° - 30° = 60°.
5. In Triangle AOC:
- In triangle AOC, ∠OAC = 30° (since AO is the radius and AC is the diagonal).
- ∠OAC and ∠ACD are corresponding angles, so they are equal.
- ∠OAC = ∠ACD = 60°.
6. Using Trigonometry:
- In triangle AOC, we have a right-angled triangle with ∠OAC = 60°.
- Let BC = x. Then, AC = 2x (as AC = BD).
- In triangle AOC, we have: tan 60° = x / AO = √3.
- Therefore, x = √3 * AO.
7. Conclusion:
- From step 6, we have BC = √3 * AO.
- Therefore, AO = BC / √3.
- Hence, the radius of the circle is equal to BC. So, the correct answer is option 'B'.
- Rectangle ABCD is inscribed in a circle with center O.
- AC is the diagonal of the rectangle.
- ∠BAC = 30°.
Solution:
1. Properties of Inscribed Angle:
- In a circle, if a quadrilateral is inscribed, then the opposite angles of the quadrilateral are supplementary.
- So, ∠BAD + ∠BCD = 180°.
2. Diagonals of Rectangle:
- In a rectangle, the diagonals are equal in length and bisect each other.
- So, AC = BD.
3. In a Rectangle:
- In a rectangle, the diagonals are perpendicular bisectors of each other.
- ∠BAC = 30° implies ∠BAD = 60°.
- In triangle ABD, ∠ADB = 90°.
- So, ∠BCD = 90° - ∠BAD = 90° - 60° = 30°.
4. In Triangle ACD:
- In triangle ACD, ∠CAD = 90°.
- So, ∠CAD + ∠ACD = 90°.
- ∠ACD = 90° - ∠CAD = 90° - 30° = 60°.
5. In Triangle AOC:
- In triangle AOC, ∠OAC = 30° (since AO is the radius and AC is the diagonal).
- ∠OAC and ∠ACD are corresponding angles, so they are equal.
- ∠OAC = ∠ACD = 60°.
6. Using Trigonometry:
- In triangle AOC, we have a right-angled triangle with ∠OAC = 60°.
- Let BC = x. Then, AC = 2x (as AC = BD).
- In triangle AOC, we have: tan 60° = x / AO = √3.
- Therefore, x = √3 * AO.
7. Conclusion:
- From step 6, we have BC = √3 * AO.
- Therefore, AO = BC / √3.
- Hence, the radius of the circle is equal to BC. So, the correct answer is option 'B'.
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A rectangle ABCD is inscribed in a circle with centre O. If AC is the diagonal and ∠BAC = 30°, then the radius of the circle will be equal to:a)(√3/2)BCb)BCc)√3BCd)2 BCCorrect answer is option 'B'. Can you explain this answer?
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