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# In the figure shown, quadrilateral ABCD is inscribed in a circle of

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Math Expert
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In the figure shown, quadrilateral ABCD is inscribed in a circle of [#permalink]

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03 Jul 2016, 10:51
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In the figure shown, quadrilateral ABCD is inscribed in a circle of radius 5. What is the perimeter of quadrilateral ABCD?

(1) The length of AB is 6 and the length of CD is 8.
(2) AC is a diameter of the circle.

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2016-07-03_2149.png [ 15.81 KiB | Viewed 9667 times ]
[Reveal] Spoiler: OA

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In the figure shown, quadrilateral ABCD is inscribed in a circle of [#permalink]

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03 Jul 2016, 11:11
2
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Perimeter of ABCD = AB + BC + CD + AD = ?

St1: The length of AB is 6 and the length of CD is 8. --> AB = 6 and CD = 8
Not Sufficient as value of BC and AD is not known.

St2: AC = diameter = 10 --> ABC and ADC are right angled triangles. Values of the sides of the quadrilateral cannot be derived.
Insufficient

Combining St1 and St2: AB = 6, CD = 8, ABC and ADC are right angled triangles
In ABC, AC = 10, AB = 6 --> BC = sqrt(10^2 - 6^2) = 8
In ADC, AC = 10, CD = 8 --> AD = sqrt(10^2 - 8^2) = 6
Perimeter = AB + BC + CD + AD = 28
Sufficient

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Senior Manager
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In the figure shown, quadrilateral ABCD is inscribed in a circle of [#permalink]

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03 Jul 2016, 23:29
we need 2 imp info-

1)ac is diamter
2)ac is perpendicular to BD

combining both we don't know AC is perpndclr to BD

SO C

we can find the lengths from pythagorus theorem.

Last edited by hsbinfy on 04 Jul 2016, 20:25, edited 1 time in total.

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In the figure shown, quadrilateral ABCD is inscribed in a circle of [#permalink]

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03 Jul 2016, 23:40
1
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Vyshak wrote:
Perimeter of ABCD = AB + BC + CD + AD = ?

St1: The length of AB is 6 and the length of CD is 8. --> AB = 6 and CD = 8
Not Sufficient as value of BC and AD is not known.

St2: AC = diameter --> We can infer that AC bisects line segment BD
Let O be the point of intersection of AC and BD.
Triangles AOD and AOB are congruent --> Therefore AB = AD
Triangles COB and COD are congruent --> Therefore BC = CD

AC = 10
Not Sufficient to determine the perimeter

Combining St1 and St2: Perimeter = AB + BC + CD + AD = 2(AB + CD) = 2(6 + 8) = 28
Sufficient

St2: AC = diameter --> We can infer that AC bisects line segment BD

how can you infer above statement until it is given AC is perpendicular to BD?

it will only bisect if AC is perpendicular to BD OR AC and BD intersect at centre of circle

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Math Forum Moderator
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Re: In the figure shown, quadrilateral ABCD is inscribed in a circle of [#permalink]

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04 Jul 2016, 00:33
hsbinfy wrote:
Vyshak wrote:
Perimeter of ABCD = AB + BC + CD + AD = ?

St1: The length of AB is 6 and the length of CD is 8. --> AB = 6 and CD = 8
Not Sufficient as value of BC and AD is not known.

St2: AC = diameter --> We can infer that AC bisects line segment BD
Let O be the point of intersection of AC and BD.
Triangles AOD and AOB are congruent --> Therefore AB = AD
Triangles COB and COD are congruent --> Therefore BC = CD

AC = 10
Not Sufficient to determine the perimeter

Combining St1 and St2: Perimeter = AB + BC + CD + AD = 2(AB + CD) = 2(6 + 8) = 28
Sufficient

St2: AC = diameter --> We can infer that AC bisects line segment BD

how can you infer above statement until it is given AC is perpendicular to BD?

it will only bisect if AC is perpendicular to BD OR AC and BD intersect at centre of circle

Yes we cannot infer from Statement 2 that the two triangles are congruent. Thanks for spotting it.

But the answer is still C.

Since AC is the diameter, ABC and ADC are right angled.

In ABC, AC = 10, AB = 6 --> BC = sqrt(10^2 - 6^2) = 8
In ADC, AC = 10, CD = 8 --> AD = sqrt(10^2 - 8^2) = 6

Perimeter of the triangle = 6 + 8 + 6 + 8 = 28.

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Re: In the figure shown, quadrilateral ABCD is inscribed in a circle of [#permalink]

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10 Jul 2016, 07:36
1
KUDOS
Given: Radius of Circle = 5
Perimeter of ABCD AB + BC + CD + AD = ?

Statement 1: The length of AB is 6 and the length of CD is 8. --> AB = 6 and CD = 8
Not Sufficient, since the value of BC and AD is not known.

Statement 2: AC = diameter = 10 --> ABC and ADC are right angled triangles. Values of the sides of the quadrilateral cannot be derived.
Insufficient

Combining St1 and St2: AB = 6, CD = 8, ABC and ADC are right angled triangles
Perimeter can be derived.

Sample Calculation:
ABC right angle triangle, AC =10, AB = 6 $$BC^2 = AC^2 - AB^2$$
BC = 8

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In the figure shown, quadrilateral ABCD is inscribed in a circle of [#permalink]

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10 Jul 2016, 19:04
1
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Top Contributor

In the figure shown, quadrilateral ABCD is inscribed in a circle of radius 5. What is the perimeter of quadrilateral ABCD?

(1) The length of AB is 6 and the length of CD is 8......>no information about the lengths of other two sides(information required because we don't know which specific quadrilateral is quadrilateral ABCD above),Insufficient

(2) AC is a diameter of the circle.........>So now we know AC=10 and its opposite angle $$\angle$$ABC=$$90^{\circ}$$(for $$\triangle$$ ABC) and $$\angle$$ADC=$$90^{\circ}$$(for $$\triangle$$ ADC),But still we don know the lengths of any sides of the Quadrilateral ABCD,Insufficient

(1)+(2) AB=6,AC=10 and $$\angle$$ABC=$$90^{\circ}$$ so According to pythagorean triple BC=$$\sqrt{AC^2+AB^2}$$=$$\sqrt{10^2-6^2}$$=8

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Re: In the figure shown, quadrilateral ABCD is inscribed in a circle of [#permalink]

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27 Apr 2017, 15:40
each alone is not sufficient.
1+2 => we have 2 right triangles (properties of inscribed triangles in a circle, where the hypotenuse is the diameter).
we know 2 sides, we can find 3rd side for each triangle.
in the end, we can find the perimeter.

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Re: In the figure shown, quadrilateral ABCD is inscribed in a circle of [#permalink]

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29 Sep 2017, 05:47
Hi,

And one of the properties of a cyclic quadrilateral is AB + CD = AD + BC (based on given fig.)

So shouldn't A be the answer choice?

Can someone explain pl.

Thanks

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Math Expert
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Re: In the figure shown, quadrilateral ABCD is inscribed in a circle of [#permalink]

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29 Sep 2017, 06:14
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Expert's post
Ashokshiva wrote:
Hi,

And one of the properties of a cyclic quadrilateral is AB + CD = AD + BC (based on given fig.)

So shouldn't A be the answer choice?

Can someone explain pl.

Thanks

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Re: In the figure shown, quadrilateral ABCD is inscribed in a circle of   [#permalink] 29 Sep 2017, 06:14
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