Jean Farrell

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 $\overline{\mathit{A}\mathit{D}}$ the diameter of the circle?

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Chloe Barr

Expert

Step 1
According to the Inscribed-Angle Theorem, $\mathrm{\angle }\mathit{A}\mathit{B}\mathit{C}\cong \mathrm{\angle }\mathit{D}\mathit{E}\mathit{F}$, and, according to the Side-Angle-Side Theorem, $\mathrm{△}\mathit{A}\mathit{B}\mathit{C}\cong \mathrm{△}\mathit{D}\mathit{E}\mathit{F}$. In particular, $\mathrm{\angle }\mathit{B}\mathit{A}\mathit{C}$ $\cong \mathrm{\angle }\mathit{E}\mathit{D}\mathit{F}$, and $\overline{\mathit{A}\mathit{C}}\cong \overline{\mathit{D}\mathit{F}}$. Again, according to the Inscribed-Angle Theorem, $\mathrm{\angle }\mathit{A}\mathit{D}\mathit{C}\cong \mathrm{\angle }\mathit{D}\mathit{A}\mathit{F}$.
Likewise, $\mathrm{\angle }\mathit{B}\mathit{A}\mathit{F}\cong \mathrm{\angle }\mathit{C}\mathit{D}\mathit{E}$, and $\overline{\mathit{B}\mathit{F}}\cong \overline{\mathit{C}\mathit{E}}$.
Since $\mathrm{m}\mathrm{\angle }\mathit{B}\mathit{A}\mathit{C}+\mathrm{m}\mathrm{\angle }\mathit{C}\mathit{A}\mathit{D}+\mathrm{m}\mathrm{\angle }\mathit{D}\mathit{A}\mathit{F}=\mathrm{m}\mathrm{\angle }\mathit{B}\mathit{A}\mathit{F}=\mathrm{m}\mathrm{\angle }\mathit{C}\mathit{D}\mathit{E}=\mathrm{m}\mathrm{\angle }\mathit{E}\mathit{D}\mathit{F}+\mathrm{m}\mathrm{\angle }\mathit{F}\mathit{D}\mathit{A}+\mathrm{m}\mathrm{\angle }\mathit{A}\mathit{D}\mathit{C},$, $\mathrm{m}\mathrm{\angle }\mathit{C}\mathit{A}\mathit{D}=\mathrm{m}\mathrm{\angle }\mathit{F}\mathit{D}\mathit{A}$
Step 2
According to the Angle-Side-Angle Theorem, $\mathrm{△}\mathit{A}\mathit{D}\mathit{F}\cong \mathrm{△}\mathit{D}\mathit{A}\mathit{C}$, in particular, $\mathrm{\angle }\mathit{A}\mathit{F}\mathit{D}\cong \mathrm{\angle }\mathit{D}\mathit{C}\mathit{A}$. These angles are supplementary, though. Consequently, they are right angles. According to the Inscribed-Angle Theorem, $\overline{\mathit{A}\mathit{D}}$ is a diameter of the circle.

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Ryland Houston

Expert

Explanation:
The lengths of the edges $\overline{ABCDEFA}$ go 2, 7, 11, 2, 7, 11 which is cyclic of order 2, so the inscribed hexagon has rotational symmetry if turned through $\frac{2\pi }{2}$. It follows immediately that any diagonal drawn between pairs of opposite vertices is a perfect diameter.

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