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Quadrilateral ABCD is inscribed in a circle with angle BAC = 70°, angl

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Quadrilateral ABCD is inscribed in a circle with angle BAC = 70°, angl  [#permalink]

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New post 25 Apr 2019, 02:15
1
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A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

52% (03:11) correct 48% (02:40) wrong based on 23 sessions

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Re: Quadrilateral ABCD is inscribed in a circle with angle BAC = 70°, angl  [#permalink]

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New post 25 Apr 2019, 17:34
2
IMO B


if we draw the quadrilateral ,


amd calculate the angles ( especially using the property that angles in se segment by any chord are equal )

we can see that the triangle ABC will become an isoselece triangle with angles 70,70,40


so this way we can say that BC = AC = 6


award kudos if helpful

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Re: Quadrilateral ABCD is inscribed in a circle with angle BAC = 70°, angl  [#permalink]

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New post 09 Feb 2020, 23:43
Bunuel wrote:
Quadrilateral ABCD is inscribed in a circle with angle BAC = 70°, angle ADB = 40°, AD = 4, and BC = 6. What is AC?


(A) \(3 + \sqrt{5}\)

(B) 6

(C) \(\frac{9\sqrt{2}}{2}\)

(D) \(8 - \sqrt{2}\)

(E) 7


Solution:
This question is based on the following property. Once, we understand what property to use. This question is not that difficult to solve.
    • Inscribed angles subtended by the same arc of a circle are equal.
Image
Keeping this property in mind. Let's solve this question.
    • \(∠BAC = ∠BDC = 70\) [angle subtended by same arc BC]
    • \(∠ACB=∠ADB= 40\) [angle subtended by same arc AB]
Image
    In triangle ABC,
    • \(∠ABC + ∠ACB+∠BAC = 180\)
    • \(∠ABC= 180 – 70 – 40\)
    • \(∠ABC = 70\)
    • \(∠ABC = ∠BAC = 70\)
Therefore, triangle ABC is an isosceles triangle.
    • AC = BC
    • BC = 6
      o AC = 6
    .
Hence, the correct answer is Option B.
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Re: Quadrilateral ABCD is inscribed in a circle with angle BAC = 70°, angl   [#permalink] 09 Feb 2020, 23:43
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