Radius of Circle: Solving SAT Practice Problem

[lax1113]

Homework Statement

Two points, A and B are on a circle, with point O being the center. The lin segment connecting A and B is equal to 2, and the perpendicular distance from the line segment AB to the center is equal to 1. What is the radius of the circle?

The Attempt at a Solution

I think i might have this one, I can draw it out and it seems to work, but I am not positive why.
So I have the perpendicular being 1, and since its a circle, it seems to me that on each side of that perpendicular point, the two parts will be equal. Now I have a right triangle, sides of 1, 1, and then the hypotenuse. So 1 squared plus 1 squared is equal to 2... to get the hypotenuse, in this case, the radius, it would be the squareroot of 2. Correct?

 

[Avodyne]

Yes, that's correct.

 

[Hurkyl]

I think i might have this one, I can draw it out and it seems to work, but I am not positive why.

What part don't you understand?

 

[lax1113]

Hurkyl,
My solution at it was kind of a guess, because I didnt know if the measure of the perpendicular was going to be in the center of the line segment or not. The fact that I could draw it so many different ways, but still it would essentially be the same, got me thinking that maybe it was only square root of 2 when i drew it that particular way.

Thank you for checking it out though, both of you.

 

[Hurkyl]

Hurkyl,
My solution at it was kind of a guess, because I didnt know if the measure of the perpendicular was going to be in the center of the line segment or not.

It's not an unreasonable guess -- so the question, now, is to see if you can prove whether or not it's actually true. Have you tried applying your knowledge of geometry to the problem? In particular, facts about triangles, perpendiculars, and symmetries might be relevant...


In a real multiple-choice test-taking environment, if you feel very confident about your guess, a reasonable thing to do would be to base your initial answer on this guess, and move on. And then once you're through the set of questions, you can come back to the problem and spend more time on it to make sure. (You might want to devote a section of your scratch paper to contain a list of problems you want to work more on after your first time through the problems)

 

[lax1113]

Hurkyl,
One thing about this SAT practice deal was that it was in class, and no scratch paper was given. (I haven't taken SAT's since last year so i can't remember, but supposedly no scratch paper is allowed, just writing on the test.) Anyway, I think my big prolem was that i wasn't SURE what a perpendicular distance was. The way math is taught at my school is nothing short of idiotic. We have this system called IMP, its supposed to be math in context, rather than just learning the basics for now. Because of this, a lot of stuff gets left out until 11th and 12th grade (im in 12th now) when it gets back to calculus and algebra based instead of working 100% out of word problems. I assumed that perpendicular distance was the most obvious thing, and then since its a circle, the perp would have to be an equal bisector of the segment, but that is where i met some kind of doubt. I really am looking to get as close to 800 on my math as possible (last time was 720), so even though its only a small part, and most likely WONT show up on my test, I want to be sure that i know how.