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

Question

https://gmatclub.com/forum/download/file.php?id=30869 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. 2016-07-03_2149.png

Kurtosis · Answer

Given: Radius = 5
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

Answer: C

Badari · Answer

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

Answer: C

AbdurRakib · Answer

https://gmatclub.com/forum/download/file.php?id=30869 
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  	riangle  ABC) and  \angle ADC= 90^{\circ} (for  	riangle  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

Accordingly we can find AD=6

So,perimeter of quadrilateral ABCD=AB+BC+CD+AD=6+8+8+6=28, Sufficient

Correct Answer C

Roosterbooster · Answer

(1) is very easily insuff but...

If we know that AC is the diameter of the circle and thus = 10, why wouldn't we say suff for (2) just off of the Pythagorean triplet 6-8-10?

That would tell me that AB/AD = 6 and BC/DC = 8.

6+6+8+8 = 28

amanvermagmat · Answer

Hi

Based on statement 2, yes we know that both ABC and ADC are right angle triangles with hypotenuse AC=10. But we cannot just assume that a right angle triangle with hypotenuse 10 will only be a 6-8-10 triangle. It could also be 5√2-5√2-10 triangle OR it could be 2√5-4√5-10 triangle.

Only if we are given that its a right angle triangle with hypotenuse 10 and one side 6 (or 8), then we can conclude that the third side would be 8 (or 6). Or if we are given that all sides are integers and hypotenuse is 10, can we conclude that the other two sides (legs) would be 6/8.

hibobotamuss · Answer

Hello,

Wondering we need 2 at all? 
Doesn't inscribed mean that the points touch the ends of the circle? Which means its the diameter anyway??

amanvermagmat · Answer

Hello

Yes, inscribed means that all the points touch the ends of the circle. But that doesnt necessarily mean that AC is diameter only. 
AC and BD are the diagonals of this quadrilateral, but its quite possible that none of those two diagonals pass through the center of the circle.

Thats why second statement is needed.

altairahmad · Answer

Bunuel   chetan2u

Does the condition AC = diameter automatically mean that AB = AD and BC = DC, or these triangles can have different values of their sides and still be right triangles ?

chetan2u · Answer

No, AC as diameter does not mean AB = AD and BC = DC.
B can be anywhere on the arc ABC, and similarly D can be anywhere on arc ADC.

ScottTargetTestPrep · Answer

Solution:

Question Stem Analysis:
 
We need to determine the perimeter of quadrilateral ABCD, which is inscribed in a circle of radius 5.

Statement One Alone:

We are given the length of two sides of the quadrilateral; however, we can’t determine its perimeter without knowing the length of the other two sides. Statement one alone is not sufficient.

Statement Two Alone:

Knowing AC is a diameter of the circle does not allow us to determine its perimeter since we don’t know the length of any sides of the quadrilateral. Statement two alone is not sufficient.

Statements One and Two Together:

Since AC is a diameter of the circle and the radius of the circle is 5, AC = 10. Furthermore, triangles ABC and ADC are right triangles with AC being the hypotenuse. Therefore, BC = 8 since AB = 6 and AC = 10. Similarly, AD = 6 since CD = 8 and AC = 10. So, the perimeter of quadrilateral ABCD is AB + BC + CD + AD = 6 + 8 + 8 + 6 = 28. Both statements together are sufficient.

Answer: C

yipmewmew · Answer

I dont understand how just by statement 2 everyone is inferring that ADC and ABC are right angle triangles. how do we know that this quad is a rectangle? why not something like a //gm?
Kindly help

chetan2u · Answer

AC is the diameter. So, B could be anywhere on the circumference, the angle ABC will ALWAYS be 90.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards