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Problem 330

If two chords of a circle subtend different acute inscribed angles, the smaller angle belongs to the shorter chord. Prove the statement and its converse.

Expert verified

The proof has two parts:
1. For the given statement: If two chords of a circle subtend different acute inscribed angles, the smaller angle belongs to the shorter chord, use the inscribed angle theorem to find measures of arcs AB and CD. Since ∠ACB < ∠ADB, we get m(arc AB) < m(arc CD). The length of a chord is proportional to the length of its intercepted arc, so AB < CD.
2. For the converse: If AB < CD, then using the properties of chords and arcs, we have m(arc AB) < m(arc CD). Using the inscribed angle theorem, we get ∠ACB < ∠ADB. This completes the proof of the statement and its converse.

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Chapter 17

Given: \(\underline{A B}\) and \(\underline{C D}\) are chords of $\odot P ; \underline{A B} \cong \underline{C D}\(. Prove: \)\underline{\mathrm{AC}} \cong \mathrm{BD}^{-}$

Chapter 17

Show that in the same circle (or in congruent circles) if two chords are not congruent, then the longer chord is nearer the center of the circle than the shorter chord.

Chapter 17

If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\), and \(\mathrm{d}\) are four equidistant parallel chords in a circle, prove that \(\mathrm{a}^{2}-\mathrm{d}^{2}=3\left(\mathrm{~b}^{2}-\mathrm{c}^{2}\right)\).

Chapter 17

An arch is built in the form of an arc of a circle and is subtended by a chord \(30 \mathrm{ft}\). long. If a chord \(17 \mathrm{ft}\). long subtends half that arc, what is the radius of the circle?

Chapter 17

In a circle $\mathrm{O}, \underline{\mathrm{OE}} \perp \underline{\mathrm{AB}}, \underline{\mathrm{OF}} \perp \mathrm{CD}$ and \(\mathrm{m} \angle \mathrm{OEF}>\mathrm{m}<\mathrm{OFE}\). Prove that \(\mathrm{AB}>\mathrm{CD}\).

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