Circle problem

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JurajJay

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Jun 26, 2019
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Hello !

I tried to solve this problem without calculator but couldn't find a way to do it. I always used trigonometry to solve it.
There is a way I found, but you need to know that sin(30°) = 0.5 beforehand.
Can you find some way to solve this without calculator ?

Thanks in advance :)

The problem: What is the distance between points B and D ?CircleProblem.jpg
 
Is there a prohibition against the sine of 30º being 1/2? Had you just asked me as we met on the street, I would have told you that - without reference to a calculator. As we are required to use no calculator at all, that should include your brain and any book that might include a table of values. Utterly silly.

Is AD a diameter?
BD Distance? Which one? Secant or Circular?

What paths have you trod?
 
tkhunny said:
Is there a prohibition against the sine of 30º being 1/2? Had you just asked me as we met on the street, I would have told you that - without reference to a calculator. As we are required to use no calculator at all, that should include your brain and any book that might include a table of values. Utterly silly.

Is AD a diameter?
BD Distance? Which one? Secant or Circular?

What paths have you trod?
Click to expand...

I know that most people probably know that sine of 30° is 0.5. But i just wanted to know if there is a different solution. Secant distance. AD is diamater The problem said no use of any tables, books or calculators, so that's why I wanted to know if there is different solution.

When I think about it, I solved it without calculator. The problem said to solve it without calculator and I did it, so I guess I don't have to try and find other way. Might delete this post, thanks for the answer (if it's even possible)
 
Last edited:
JurajJay said:
I tried to solve this problem without calculator but couldn't find a way to do it. I always used trigonometry to solve it.
There is a way I found, but you need to know that sin(30°) = 0.5 beforehand.
Can you find some way to solve this without calculator ?

The problem: What is the distance between points B and D ?
Click to expand...
It will be helpful if you can quote the words used in stating the problem; we shouldn't be left to assume that AD is a diameter, that it is CE that is 3 units long, etc.

But there is no reason you can't use trigonometry, just because you can't use a calculator.

All you'll have to do differently is to not actually determine the angles, but just find the sines and cosines of angles, using things like that addition and subtraction identities. You know the sine and cosine of 30 degrees; you can write expressions for various other sines or cosines in terms of a variable for the diameter; and so on. I can see several ways you might approach it. It sounds like you found one of them.

If you're asking whether it is legal to "know" sin(30°), of course it is. If you hadn't learned it, you could just draw in a equilateral triangle and work it out from there.
 
Dr.Peterson said:
It will be helpful if you can quote the words used in stating the problem; we shouldn't be left to assume that AD is a diameter, that it is CE that is 3 units long, etc.

But there is no reason you can't use trigonometry, just because you can't use a calculator.

All you'll have to do differently is to not actually determine the angles, but just find the sines and cosines of angles, using things like that addition and subtraction identities. You know the sine and cosine of 30 degrees; you can write expressions for various other sines or cosines in terms of a variable for the diameter; and so on. I can see several ways you might approach it. It sounds like you found one of them.

If you're asking whether it is legal to "know" sin(30°), of course it is. If you hadn't learned it, you could just draw in a equilateral triangle and work it out from there.
Click to expand...

"There is a circle with diameter AD and two of its chords AB
and AC (see figure). | ∡BAC | = 60°, BE ⊥ AC, | AB | = 24 cm, | EC |= 3 cm.
Find the distance between points B and D."

- This is the problem. I tried to solve it with mathematics that I'm expected to know (that is basic trigonometry and pythagorean theorem in this case), since this problem was intended for 10th grade students in our country. So I was a bit skeptical about just typing directly that sine of 30° is 0.5. Now that I look at it it's silly. When I was solving the problem I drew the equilateral triangle so that it would be obvious how I came to the conclusion that sine of 30° is 0.5. They didn't specify the type of distance, so I assumed it to be secant. (The correct answer is 2√3 so that is secant distance)
Thanks for your answer
 
Last edited:
JurajJay said:
"There is a circle with diameter AD and two of its chords AB
and AC (see figure). | ∡BAC | = 60°, BE ⊥ AC, | AB | = 24 cm, | EC |= 3 cm.
Find the distance between points B and D."
- This is the problem. I tried to solve it with mathematics that I'm expected to know (that is basic trigonometry and pythagorean theorem in this case), since this problem was intended for 10th grade students in our country. So I was a bit skeptical about just typing directly that sine of 30° is 0.5. Now that I look at it it's silly. When I was solving the problem I drew the equilateral triangle so that it would be obvious how I came to the conclusion that sine of 30° is 0.5. They didn't specify the type of distance, so I assumed it to be secant. (The correct answer is 2√3 so that is secant distance)
Click to expand...
One more question: what does the symbol of an arc with a dot in it in \(\displaystyle \angle BEA\) mean?
 
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