Geometry Problem

[tom2326]

I recently encountered this problem in a maths competition at school and even after days of working on it i just can't find any real way of solving it.

I've attached the problem.

I need to find the radius and the area of the shaded area

To me it really seems like it is a variable answer and has no set number, but i'm doubting that if so it would be in this quiz.

Any help????

 

[ajb]

What have you done so far?

 

The hint is to find calculate R^2 - r^2 where R is the radius of the large circle and r the radius of the small circle. Cut the chord in half using r and then relate this to R using Pythagoras. You can continue from there.

[studiot]

I think you need one more piece of infomation to solve this.

 

You need either the rise of the arc over the chord (h), or one of the radii ajb mentions R or r, or the angle subtended at the centre.

[ajb]

I think you need one more piece of infomation to solve this.

You only actually need R^2 - r^2 to solve this and you can calculate directly, you don't need R or r themselves.

[studiot]

Here is a spread sheet calculating the value of r given various values of h = R - r

 

 

 

[mathematic]

Here is a spread sheet calculating the value of r given various values of h = R - r

 

chords1.jpg

 

 

Why is this spread sheet relevant? [latex]R^2-r^2=25.[/latex]

Edited August 5, 2015 by mathematic

[Robittybob1]

Why is this spread sheet relevant? [latex]R^2-r^2=25.[/latex]

so does that mean the area is always constant irrespective of the radii values?

[studiot]

 

Why is this spread sheet relevant?

 

The question requires you to find r and R.

 

The relationship between them is h = R - r and R = h/2 + c²/8h where c is the chord length.

 

You are only given c and require one other of h, R or r to complete the calculation.

Edited August 5, 2015 by studiot

[ajb]

The question requires you to find r and R.

mathematic is right, assuming the circles are concentric and the chord is tangent to the inner circle; I think that is the case. You only need R^2 -r^2 to solve this problem. You just need a factor of pi now and the problem is solved.

[studiot]

 

post#1

 

I need to find the radius and the area of the shaded area

 

This is pretty clear to me.

 

Perhaps the OP was attempting the wrong problem.

[ajb]

I don't see how you can find the radius (of what exactly?) without extra information. Finding the area is easy enough.

Edited August 6, 2015 by ajb

[studiot]

 

I don't see how you can find the radius (of what exactly?) without extra information. Finding the area is easy enough.

 

 

Isn't that what I said in post#3

[ajb]

Isn't that what I said in post#3

Indeed, so we think the original question is ill posed. tom2326 has probabily misunderstood the question, or missed some vital information, like a value for r or R, depending on the exact question.

[studiot]

Yup.

 

My guess (after your responses) is that the original question only asked for the area, but Tom assumed you need to calculate the ( or a) radius and, of course, couldn't do that.

So he asked for the radius as well

 

I never considered that possibility in my original response.

[ajb]

Indeed, it is tempting to try to find R and r and then the areas and work out the difference. But that won't work!