Jean Farrell

Jean Farrell

Answered

2022-09-22

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

Expert

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

Expert

2022-09-24Added 4 answers

Explanation:
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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