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

Answer & Explanation

Chloe Barr

Expert

2022-09-23Added 6 answers

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, $\mathrm{\u25b3}\mathit{A}\mathit{B}\mathit{C}\cong \mathrm{\u25b3}\mathit{D}\mathit{E}\mathit{F}$. 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, $\mathrm{\u25b3}\mathit{A}\mathit{D}\mathit{F}\cong \mathrm{\u25b3}\mathit{D}\mathit{A}\mathit{C}$, 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.

Ryland Houston

Expert

2022-09-24Added 4 answers

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.

Most Popular Questions