triangle in a circle

[JH123]

The Question: Draw a triangle, with all its vertices on a circle, that meets the given conditions. If it is impossible, explain why. Conditions: An isosceles triangle with two sides that are radii of the same circle.

The book answer is as follows: Such a triangle cannot be drawn, since the intersection of the two radii--the center of the circle--is not on the circle.

?? I'm really confused. An isosceles triangle has 2 congruent sides. All radii of a circle are congruent. The center of the circle, is well, at the center of the circle. I can't figure out how or why this isn't doable. Is the book wrong?

Thanks for any help with this.

 

[burakaltr]

The Question: Draw a triangle, with all its vertices on a circle, that meets the given conditions. If it is impossible, explain why. Conditions: An isosceles triangle with two sides that are radii of the same circle.

The book answer is as follows: Such a triangle cannot be drawn, since the intersection of the two radii--the center of the circle--is not on the circle.

?? I'm really confused. An isosceles triangle has 2 congruent sides. All radii of a circle are congruent. The center of the circle, is well, at the center of the circle. I can't figure out how or why this isn't doable. Is the book wrong?

Thanks for any help with this.

Well one such triangle is formed by 45 degree 2 adjacent sides of Length "r" and the 90 degree between the two subtends 180 deg when taken to central line



(r,r,2r) Triangle is possible where 2r is hypotenuse of r,r

BUT

apply Pythagorean's Theorem

r**2 + r**2 = (2r)**2

2 r**2 = 4(r**2)

There exists NO SOLUTION !

(I wish I could have illustrated this )


ALSO

if you draw one instance of such an impossible triangle

you'll see that 2x+2y+2z=0 where it should equal 180

Draw one for yourself,

the only possibility is the Diameter where it does not denote a Triangle, actually

 

[soroban]

Hello, JH123!

Draw a triangle, with all its vertices on a circle, that meets the given conditions.
If it is impossible, explain why.
Condition: An isosceles triangle with two sides that are radii of the same circle.

The book answer is as follows: Such a triangle cannot be drawn,
since the intersection of the two radii (the center of the circle) is not on the circle.

I'm really confused.
An isosceles triangle has 2 congruent sides.
All radii of a circle are congruent.
The center of the circle, is well, at the center of the circle.

I can't figure out how or why this isn't doable.
Is the book wrong?

It says: we have a triangle with all its vertices on a circle.

It might look like this:

              * ♥ *
          *    ◊ ◊    *
        *     ◊   ◊     *
       *     ◊     ◊     *
            ◊       ◊
      *    ◊         ◊    *
      *   ◊           ◊   *
      *  ◊             ◊  *
        ◊               ◊
       ♥  ◊  ◊  ◊  ◊  ◊  ♥
        *               *
          *           *
              * * *

Then we are told that two sides of the triangle are radii of the circle.
Then the triangle must look like this.

              * * *
          *           *
        *               *
       *                 *

      *                   *
      *         ♥         *
      *      ◊     ◊      *
          ◊           ◊ 
       ♥  ◊  ◊  ◊  ◊  ◊  ♥
        *               *
          *           *
              * * *

But the second triangle must have a vertex
. . at the center of the circle, not on the circumference.

Therefore, the construction is impossible.

 

[JH123]

Thanks!!

I was so focused on two sides being radii, that the vertices being on the circumference of the circle completely escaped me. These answers were great, the illustrated answer made it easy to see where I had gone wrong. Thank you so much for taking the time to help!