Prove that if an arc of a circle subtends a right angle at any point on the remaining part of the circle, then the arc is a semi-circle.
Prove that if an arc of a circle subtends a right angle at any point on the remaining part of the circle, then the arc is a semi-circle.
Given : A circle with centre O and an arc AB subtending ∠ACB at a point C on the remaining part of the circle such that ∠ACB = 90∘.
To Prove: ArcAB is a semi-circle.
Construction : Join OA and OB
STATEMENT REASON
1. Arc AB subtends ∠AOB at the centre
and ∠ACB at a point C on the
remaining part of the circle.
∠AOB = ∠ACB .............. (i) Angle at the centre is double the angle at a point on the remaining part of the circle.
2. ∠ACB = 90∘ ..........(ii) Given
3. ∠AOB = (2 x 90∘) = 180∘ From (i) and (ii)
AOB is a straight line
AOB is a diameter Chord AB passes through the centre O.
Arc AB is a semi-circle.
Given : A circle with centre O and an arc AB subtending ∠ACB at a point C on the remaining part of the circle such that ∠ACB = 90∘.
To Prove: ArcAB is a semi-circle.
Construction : Join OA and OB
STATEMENT REASON
1. Arc AB subtends ∠AOB at the centre
and ∠ACB at a point C on the
remaining part of the circle.
∠AOB = ∠ACB .............. (i) Angle at the centre is double the angle at a point on the remaining part of the circle.
2. ∠ACB = 90∘ ..........(ii) Given
3. ∠AOB = (2 x 90∘) = 180∘ From (i) and (ii)
AOB is a straight line
AOB is a diameter Chord AB passes through the centre O.
Arc AB is a semi-circle.
