What is b in the circle shown above?

From the given data,

1) b is a common side to both the right angled triangles
2) Both the triangles have one common angle (90 degrees)

So,
c/a = b/b
=> c=a

And by using Pythagoras theorem,
84= (b*b) + (5 * 5)

Hence Statement 2 alone is sufficient.


Kudos +1 guys!

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hey, I am confused as to c=a in this case. could you explain this?

Confused as well. Detailed explanation will be helpful.

The STE 1 can be made from the figure using the theory of similar triangles. using Pythagorean theory we can make an equation with only one variable. B is sufficient

per your logic, each triangle inscribed in a circle is a isosceles right triangle. I think you are confusing it with another property or similar triangles i.e ratio of area = square of the ratio of sides

from the similar triangle we can say that b/5 = a/c or bc =5a and c^2 = b^2 + 5^2

St 1: bc=5a. Mentioned above. INSUFFICIENT
St 2: a = √84 therefore bc = 5√84 and c^2 = b^2 +25. two variable, two equation. SUFFICIENT

Option B

Could you please explain how do you know that the triangles are similar? Based on the given information, don't you need to confirm that an additional pair of sides is similar, or that an additional pair of angles is equal?

Thank you!

From the information and attached figure we can derive:

angle PQR = angle QOR = 90 degrees ---(1)
angle PRQ = angle QRO = x degrees ----(2)
From 1 and 2 and AAA similarity ===> angle QPR = angle QQR = y degrees

Therefore the triangles PQR and QOR are similar
PQ/QR = QO/OR ==> a/c = b/5 ==>bc =5a.........(3)

Statement 1 ==> Same as can be derived from given information and insufficient since various combinations exist
Statement 1 ==> From (3), a = √84 and b^2 + 5^2 = C^2 (since QOR is a right angle triangle) we can get the value of b

Please explain how do we find the value of 'b'

with a value as rt(84), we can find the value of diameter in terms of x+5 for triangle PQR and using the A value we can find length of x in terms of b and so on create a equation and by solving it we will get b x as 7 and b as rt(35).

I got it wrong but I'll provide my approach after thinking about it, to see if I can help.

Problem: DS to find b
The information provided by the problem is:
1. Two triangles with a common side and adjacent angles of 90º angle (lets name it L).
2. Triangle QRL measures: 5-b-c, angles Q,R,90º.
Looks like 30-60-90 but there is no certainty, still pitagoras can be applied.
3. One big triangle PQR that is isoceles Right triangle (because internal angles theory, the diameter shows that the angle is 90º). And is inside a circle so pitagoras is possible.
Mental note: Get a side then can apply "30-60-90, x- x√3, 2x" or Pitagoras.

Statements:
(1) bc=5a
Can't apply pitagoras or "30,60,90"
Not Sufficient -> BCE

2) PQ = √84
One side! So can apply pitagoras and "30-60-90"
Can find C!
with C then can find b with pitagoras.
Sufficient -> B

Seems like you made your mind
bc = 5√84 this is not right
You are taking bc=5a from the 1st statement and assuming that statement 2 alone is enough.

Can somebody please provide a clear answer, why B and not C?

Your entire solution is based that it's 30-60-90, triangle, which we don't know

I thought that kind o trick was more known or used...
Is a shortcut that can be used on right triangle that has the following angle measures has:
Angle measures: 90- 60 - 30
Side measures: 2x - x√3 - x

Is similar for equilateral:
Angle measures: 90- 45 - 45
Side measures: x√2 - x - x

Still the more possible approaches the better, this one is a hard one

I meant you can't know if it's 90 60 30 or 90 45 45 or any other angle combination
https://en.wikipedia.org/wiki/Thales%27 ... heorem.gif

You just know that this is right triangle. That's it
Actually there are 3 right triangles, which are similar to each other, so basically that kind of question could be solved through similarity but I don't see how that can be solved via option B only

can you please elaborate it? I'm confused a bit, much appreciated!

Dude you are right.... I assumed the angle measures

I'll call some cavalry to solve this as I can't see a proper way to crack this nut
VeritasPrepKarishma could you please help us with this?

Took a lot of time, but here's my solution.

From the diagram, say the diameter is d
b^2+25 = c^2, b^2+(d-5)^2=a^2 & d^2=a^2+c^2.

Using the 3 equations it can be resolved to c^2=5d, telling the 2 triangles are similar to one other.

Statement 1: bc=5a gives us the same info as above, unless we know one of the other sides we cannot solve for b. This is insufficient.

Statement 2: a^2=84, put this into the d^2=a^2+c^2 along with c^2=5d. These will resolve to a value for d from which one can solve for b. Sufficient

Please comment if you see any flaws in the above method or ways of making this faster.

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