I stumped, some kind of trapezoid inside a circle

[insanemathsguy]

Basically I'm given a circle of 4cm radius, and told that the lines of this trapezoid inside the circle are: a=6cm, d=6.5cm, and b+c=9.5cm
Well I've put my points A and D down, but I have no idea how to work out where B and C should go.

 

[HallsofIvy]

Your post makes no sense. You talk about "points" A and D and "lines" a and d. What is their relationship? And what is your question about this trapezoid?

 

[insanemathsguy]

The lines are a, b, c, d
The points are where those lines meet each other ie lines d and a intersect at point A, lines a and b intersect at point B, lines b and c intersect at point C, and lines c and d intersect at point D.
A, B, C, and D are the four corners of the trapezoid. This trapezoid is inscribed in a circle of radius 4cm.
I know that the lines a=6cm, d=6.5cm, and b+c=9.5cm, therefore I can draw points A and D. However I do not know the individual lengths of the lines b or c, therefore I don't know where to put points B and C.
The question is how do I find out the individual lengths of b and c so I know where to put B and C on the circle.

 

[insanemathsguy]

**** you're right!!
You wouldn't mind telling me how you came to knowing it was an isosceles trapezoid would you?
I've just found out that it was also a trick question to see if it was possible to find an answer based on that lack of information we were given.

 

[Bob Brown MSEE]

The lines are a, b, c, d
The points are where those lines meet each other ie lines d and a intersect at point A, lines a and b intersect at point B, lines b and c intersect at point C, and lines c and d intersect at point D.
A, B, C, and D are the four corners of the trapezoid. This trapezoid is inscribed in a circle of radius 4cm.
I know that the lines a=6cm, d=6.5cm, and b+c=9.5cm, therefore I can draw points A and D. However I do not know the individual lengths of the lines b or c, therefore I don't know where to put points B and C.
The question is how do I find out the individual lengths of b and c so I know where to put B and C on the circle.
---------------------------
The lines are a, b, c, d
The points are where those lines meet each other ie lines d and a intersect at point A, lines a and b intersect at point B, lines b and c intersect at point C, and lines c and d intersect at point D.
A, B, C, and D are the four corners of the trapezoid. This trapezoid is inscribed in a circle of radius 4cm.
I know that the lines a=6cm, d=6.5cm, and b+c=9.5cm, therefore I can draw points A and D. However I do not know the individual lengths of the lines b or c, therefore I don't know where to put points B and C.
The question is how do I find out the individual lengths of b and c so I know where to put B and C on the circle.

Like this?

not quite isosceles
oops! NOT B+C THIS IS C+D = 9.5, nevermind

 

[insanemathsguy]

I should really start calling my lines AB etc, I'd really save myself and I guess everyone else confusion.
So anyway, your diagram has AB=6.5cm and BC=6cm (It should be AB=6cm and DA=6.5cm so that BCD=9.5cm (I'm sure what the sides are called doesn't particularly matter but I thought it best to make clear)). So how did you get to the figures of your DA (what would be my CD) being 2.85cm and your CD (my BC) being 6.65cm? Is that just what your program pumped out or is there some calculation behind it?

Like this?
View attachment 2885
not quite isosceles
oops! NOT B+C THIS IS C+D = 9.5, nevermind

 

[wjm11]

**** you're right!!
You wouldn't mind telling me how you came to knowing it was an isosceles trapezoid would you?
I've just found out that it was also a trick question to see if it was possible to find an answer based on that lack of information we were given.

All trapezoids inscribed in circles MUST be isosceles trapezoids. Consider: The perpendicular bisectors of all chords must pass through the center of the circle. From this it can be seen that parallel chords (the two bases of the trapezoid) have a common line as their bisector. The resulting symmetry requires the sides of the the trapezoid to be equal in length. (Drawing a sketch of this may be helpful.)

In your case, two different trapezoid solutions are possible. You can have a side parallel to EITHER the side of length 6 or the side of length 6.5.

Let us test the case where we use the side of length 6 as the larger base of the trapezoid. IF the legs are length 6.5 and the total length of side + short base is 9.5, the the short base must be length 3. (Again, drawing a sketch will be helpful.)

So, if we place parallel chords of lengths 3 and 6 on a circle of radius 4, do we really get a leg length of 6.5?

The answer is no; the distance formula can be used to show the leg length would be 6.52851 cm.

To meet your criteria exactly would require a slightly smaller circle.

Please feel free to test our other case where the bases of the trapezoid are 6.5 and 3.5 in length.