In the given figure, chord AB ≅ chord CD, Prove that, arc AC ≅ arc BD.
Chapter 3 – Circle – Text Book Solution
Practice Set 3.3 | Q 3 | Page 64
In the given figure, chord AB ≅ chord CD, Prove that, arc AC ≅ arc BD.
(1) It is given that ∆QRS is an equilateral triangle.
∴ chord RS = chord QS = chord QR (Sides of an equilateral triangle are equal)
⇒ m(arc RS) = m(arc QS) = m(arc QR) .....(1) (Corresponding arcs of congruent chords of a circle are congruent)
(2) m(arc RS) + m(arc QS) + m(arc QR) = 360º (Measure of a complete circle is 360º)
⇒ m(arc RS) + m(arc RS) + m(arc RS) = 360º [Using (1)]
⇒ 3 × m(arc RS) = 360º
⇒ m(arc RS) = 120º
∴ m(arc RS) = m(arc QR) = 120º
Now,
m(arc QRS) = m(arc QR) + m(arc RS)
⇒ m(arc QRS) = 120º + 120º = 240º
Explanation:-
We are given that triangle ∆QRS is an equilateral triangle. Therefore, all three sides of the triangle are equal. Hence, chord RS, QS, and QR are equal.
From the above, we can conclude that the measure of arc RS is equal to the measure of arc QS and arc QR. This is because the corresponding arcs of congruent chords of a circle are congruent.
Let’s assume that the measure of arc RS is x. Then, from (2), we know that:
m(arc RS) + m(arc QS) + m(arc QR) = 360°
Substituting the values we know:
x + x + x = 360°
Simplifying this equation, we get:
3x = 360°
x = 120°
Therefore, the measure of arc RS, arc QS, and arc QR is 120° each.
Now, we need to find the measure of arc QRS. Using the formula:
m(arc QRS) = m(arc QR) + m(arc RS)
Substituting the values, we get:
m(arc QRS) = 120° + 120°
m(arc QRS) = 240°
Thus, the measure of arc QRS is 240°.
Chapter 3 – Circle – Text Book Solution
Practice set 3.3 | Q 3 | Page 64
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