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Dovid
Member since Aug-6-04

Apr-04-07, 02:02 PM (EST)

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"Ptolemy's theorem - cannot figure out how to apply"

I am a retired high school math teacher.

I am tutoring a student, but can't for the life of me know where to begin on this problem:

A triangle inscribed in a circle of radius 5 has two sides measuring 5 and 6 respectively. Find the measure of the third side of the triangle. (Hint: You can only used Ptolemy's theorem to solve this problem.)

Thanks.

Peace!

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Subject	Author	Message Date	ID
Ptolemy's theorem - cannot figure out how to apply	Dovid	Apr-04-07	TOP
RE: Ptolemy's theorem - cannot figure out how to apply	alexb	Apr-04-07	1
RE: Ptolemy's theorem - cannot figure out how to apply	jay_shark	Apr-06-07	2
RE: Ptolemy's theorem - cannot figure out how to apply	Dovid	Apr-08-07	4
RE: Ptolemy's theorem - cannot figure out how to apply	Dovid	Apr-08-07	3
RE: Ptolemy's theorem - cannot figure out how to apply	mpdlc	Apr-09-07	5
RE: Ptolemy's theorem - cannot figure out how to apply	mpdlc	Apr-09-07	6

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alexb
Charter Member
1983 posts

Apr-04-07, 02:11 PM (EST)

1. "RE: Ptolemy's theorem - cannot figure out how to apply"
In response to message #0

>A triangle inscribed in a circle of radius 5 has two sides
>measuring 5 and 6 respectively. Find the measure of the
>third side of the triangle. (Hint: You can only used
>Ptolemy's theorem to solve this problem.)

Can one also use the Pythagorean theorem?

Assume O is center and points A, B, C on the circle so that AB = 5, BC = 6. Let BD be a diameter. Then triangles BCD and BAD are right and in each two sides are given. You can find the third sides from the Pythagorean theorem. Apply now Ptolemy's theorem to the quadrilateral ABCD. Of the six line segments involved, 5 are known whereas the sixth is the missing side AC.

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jay_shark
Member since Mar-20-07

Apr-06-07, 11:28 PM (EST)

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2. "RE: Ptolemy's theorem - cannot figure out how to apply"
In response to message #1

First you should note that you should get two different answers .

Here is the generalization of the above . Given b and c and some fixed radius R , the length of the third side is :

a= /(2R)

For this question , if you work it out like Alex has suggested you should get two answers ; 3sqrt3 - 4 or 3sqrt3 + 4 .

Here is an alternate solution that doesn't require Ptolemy's theorem.

Draw ABC such that AB=5 and AC =6 .Let O be the center of the circle with radius 5 . Consider the first case where <BAC is obtuse .

We know that AOB is equilateral and so <ACB =30 since <AOB =60 . Now pick a point D on the line CB such that AD is perpendicular to CB . So triangle ADC is a 60 , 90 , 30 triangle. This means AD/AC =1/2 which means AD=3 . Similarly CD/AD =sqrt3 and so CD=3sqrt3 . Also triangle ADB is a 3,4,5 triangle so DB =4 . Now just add and you get BC=3sqrt3 +4 .

If you work out the other case where <BAC is acute , you should get BC' =3sqrt3-4 .

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Dovid
Member since Aug-6-04

Apr-08-07, 05:49 PM (EST)

4. "RE: Ptolemy's theorem - cannot figure out how to apply"
In response to message #2

Thank you, also.

>First you should note that you should get two different
>answers .
>
>Here is the generalization of the above . Given b and c and
>some fixed radius R , the length of the third side is :
>
>a= /(2R)
>
>For this question , if you work it out like Alex has
>suggested you should get two answers ; 3sqrt3 - 4 or 3sqrt3
>+ 4 .
>
>
>Here is an alternate solution that doesn't require Ptolemy's
>theorem.
>
>Draw ABC such that AB=5 and AC =6 .Let O be the center of
>the circle with radius 5 . Consider the first case where
><BAC is obtuse .
>
>We know that AOB is equilateral and so <ACB =30 since <AOB
>=60 . Now pick a point D on the line CB such that AD is
>perpendicular to CB . So triangle ADC is a 60 , 90 , 30
>triangle. This means AD/AC =1/2 which means AD=3 . Similarly
>CD/AD =sqrt3 and so CD=3sqrt3 . Also triangle ADB is a 3,4,5
>triangle so DB =4 . Now just add and you get BC=3sqrt3 +4 .
>
>If you work out the other case where <BAC is acute , you
>should get BC' =3sqrt3-4 .

Peace!

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Dovid
Member since Aug-6-04

Apr-08-07, 05:49 PM (EST)

3. "RE: Ptolemy's theorem - cannot figure out how to apply"
In response to message #0

Thank you so much.

Peace!

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mpdlc
Member since Mar-12-07

Apr-09-07, 11:05 AM (EST)

5. "RE: Ptolemy's theorem - cannot figure out how to apply"
In response to message #0

I think the attached solution is really easy

Calling the radius of the given circle R and the three sides of the triangle 2a, 2b, and 2c and applying Ptolomey to the quadrilateral OMaCMb where Ma and Mb are the midpoints of the original sides.

It is obvious

c = 2 ( a * SQRT (R^2 �b^2) + b * SQRT (R^2 �a^2) )/ R

by substitution R=5 ; b = 5/2 ; a = 6/2 we get c = 91.92

mpdlc

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