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

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

(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

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.

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??

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.

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 ?

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.

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

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

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

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