In fig 3.38 \(\triangle QRS\) is an equilateral triangle. Prove that,
(1) arc RS ≅ arc QS ≅ arc QR (2) m(arc QRS) = \(240^\circ\).
Figure: \(\triangle QRS\) is inscribed in a circle.
Answer: Congruent chords give congruent arcs, so each minor arc is \(120^\circ\) and arc QRS = \(240^\circ\).
(1) \(\triangle QRS\) is equilateral, so chord RS ≅ chord QS ≅ chord QR.
Arcs corresponding to congruent chords of a circle are congruent, so arc RS ≅ arc QS ≅ arc QR (the minor arcs).
(2) These three arcs together make the complete circle and have equal measures:
\[3 \times m(\text{arc QR}) = 360^\circ\] \[m(\text{arc QR}) = m(\text{arc RS}) = 120^\circ\]Arcs QR and RS have only R in common, so
\[m(\text{arc QRS}) = 120^\circ + 120^\circ = 240^\circ\]Log in or create a free account to answer this question.
