Points A, B, C are on a circle, such that m(arc AB) = m(arc BC) = 120°. No point, except point B, is common to the arcs. Which is the type of ∆ABC? Equilateral triangle Scalene triangle Right angled triangle Isosceles triangle
Chapter 3 – Circle – Text Book Solution
Problem Set 3 | Q 1.09 | Page 83
Four alternative answers for the following question is given. Choose the correct alternative.
Points A, B, C are on a circle, such that m(arc AB) = m(arc BC) = 120°. No point, except point B, is common to the arcs. Which is the type of ∆ABC?
- Equilateral triangle
- Scalene triangle
- Right angled triangle
- Isosceles triangle
>m(arc AB) = m(arc BC) = 120º
Now,
m(arc AB) + m(arc BC) + m(arc CA) = 360º
⇒ 120º + 120º + m(arc CA) = 360º
⇒ 240º + m(arc CA) = 360º
⇒ m(arc CA) = 360º − 240º = 120º
∴ m(arc AB) = m(arc BC) = m(arc CA)
⇒ arc AB ≅ arc BC ≅ arc CA ......(Two arcs are congruent if their measures are equal)
⇒ chord AB ≅ chord BC ≅ chord CA ......(Chords corresponding to congruent arcs of a circle are congruent)
∴ ∆ABC is an equilateral triangle. ......(All sides of equilateral triangle are equal)
Hence, the correct answer is option Equilateral triangle.
Explanation:-
An equilateral triangle is formed by the chords of the circle. We know that the measure of arc AB is equal to the measure of arc BC, which is equal to 120 degrees. Using the fact that the sum of the measures of the three arcs that make up a circle is equal to 360 degrees, we can find that the measure of arc CA is also 120 degrees. Therefore, all three arcs have the same measure, and by extension, all three corresponding chords (AB, BC, and CA) are congruent. Since an equilateral triangle is a triangle in which all sides are equal, we can conclude that triangle ABC is equilateral. Therefore, the correct answer is “Equilateral triangle”.
Chapter 3 – Circle – Text Book Solution
Problem Set 3 | Q 1.09 | Page 83
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