Practice Set 3.3
In figure 3.37, points G, D, E, F are concyclic points of a circle with centre C.∠ ECF…
In fig 3.38 ∆ QRS is an equilateral triangle. Prove that,(1) arc RS ≅ arc QS ≅ arc…
In fig 3.39 chord AB ≅ chord CD, Prove that, arc AC ≅ arc BD
Practice Set 3.3
Question 1.
In figure 3.37, points G, D, E, F are concyclic points of a circle with centre C.
∠ ECF = 70°, m(arc DGF) = 200° find m(arc DE) and m(arc DEF).
Answer:
Given ∠ECF = 70° and m(arc DGF) = 200°
We know that measure of major arc = 360° - measure of minor arc
m(arc DGF) = 360° - m(arc DF)
⇒ m(arc DF) = 360° - 200° = 160°
⇒∠ DCF = 160°
∵ The measure of a minor arc is the measure of its central angle.
∴m(arc DEF) = 160°
So, ∠DCE = ∠ DCF -∠ECF = 160° - 70°
⇒ ∠DCE = 90°
The measure of a minor arc is the measure of its central angle.
m(arc DE) = 90°
Question 2.
In fig 3.38 ∆ QRS is an equilateral triangle. Prove that,
(1) arc RS ≅ arc QS ≅ arc QR
(2) m(arc QRS) = 240°.
Answer:
(1) Two arcs are congruent if their measures and radii are equal.
∵∆ QRS is an equilateral triangle
∴ RS = QS = QR
⇒arc RS ≅ arc QS ≅ arc QR
(2) Let O be the centre of the circle.
m(arc QS) = ∠ QOS
∠ QOS + ∠ QOR + ∠ SOR = 360°
⇒ 3∠ QOS = 360° {∵ ∆QRS is an equilateral triangle}
⇒∠ QOS = 120°
m(arc QS) = 120°
m(arc QRS ) = 360° - 120° {∵Measure of a major arc = 360°- measure of its corresponding minor arc}
⇒m(arc QRS ) = 240°
Question 3.
In fig 3.39 chord AB ≅ chord CD, Prove that, arc AC ≅ arc BD
Answer:
∵ chord AB ≅ chord CD
∴ m(arc AB) = m(arc CD){Corresponding arcs of congruent chords of a circle (or congruent circles) are congruent}
Subtract m(arc CB) from above,
m(arc AB)– m(arc CB) = m(arc CD) – m(arc CB)
⇒m(arc AC) = m(arc BD)
⇒arc AC ≅ arc BD
Hence, proved.
