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

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Math Expert
Joined: 02 Sep 2009
Posts: 62676
Quadrilateral ABCD is inscribed in a circle with angle BAC = 70°, angl  [#permalink]

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25 Apr 2019, 02:15
1
9
00:00

Difficulty:

65% (hard)

Question Stats:

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

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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

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

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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

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

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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.

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]

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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