Jean Farrell

Jean Farrell



Hexagon ABCDEF has sides AB and DE of length 2, sides BC and EF of length 7, and sides CD and AF of length 11, and it is inscribed in a circle. Compute the diameter of the circle.
Why is A D ¯ the diameter of the circle?

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Answer & Explanation

Chloe Barr

Chloe Barr


2022-09-23Added 6 answers

Step 1
According to the Inscribed-Angle Theorem, A B C D E F , and, according to the Side-Angle-Side Theorem, A B C D E F . In particular, B A C E D F , and A C ¯ D F ¯ . Again, according to the Inscribed-Angle Theorem, A D C D A F .
Likewise, B A F C D E , and B F ¯ C E ¯ .
Since m B A C + m C A D + m D A F = m B A F = m C D E = m E D F + m F D A + m A D C ,, m C A D = m F D A
Step 2
According to the Angle-Side-Angle Theorem, A D F D A C , in particular, A F D D C A . These angles are supplementary, though. Consequently, they are right angles. According to the Inscribed-Angle Theorem, A D ¯ is a diameter of the circle.

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Ryland Houston

Ryland Houston


2022-09-24Added 4 answers

The lengths of the edges A B C D E F A ¯ go 2, 7, 11, 2, 7, 11 which is cyclic of order 2, so the inscribed hexagon has rotational symmetry if turned through 2 π 2 . It follows immediately that any diagonal drawn between pairs of opposite vertices is a perfect diameter.

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