Advertisements
Advertisements
प्रश्न
The circumcentre of the triangle ABC is O. Prove that ∠OBC + ∠BAC = 90º.
Advertisements
उत्तर
Let ABC be the triangle whose circumcenter is O.
∠OBC = ∠OCB = θ ...(Opposite angles of equal sides)
In ΔBOC, using the angle sum property of tringle, sum of all angles is 180°, we have:
∠BOC + ∠OBC + ∠OCB = 180°
⇒ ∠BOC + θ + θ = 180°
⇒ ∠BOC = 180° – 2θ
Also, in a circle, angle subtended by an arc at the center is twice the angle subtended by it at any other point in the remaining part of the circle.
∠BOC = 2∠BAC
⇒ ∠BAC = `1/2`(∠BOC)
⇒ ∠BAC = `1/2`(180° – 2θ)
⇒ ∠BAC = (90° – θ)
⇒ ∠BAC + θ = 90°
⇒ ∠BAC + ∠OBC = 90°
Hence proved.
APPEARS IN
संबंधित प्रश्न
Write True or False. Give reason for your answer.
A circle has only finite number of equal chords.
A chord PQ of a circle of radius 10 cm substends an angle of 60° at the centre of circle. Find the area of major and minor segments of the circle.
Find the length of a tangent drawn to a circle with radius 5cm, from a point 13 cm from the center of the circle.
Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at center.
If ΔABC is isosceles with AB = AC and C (0, 2) is the in circle of the ΔABC touching BC at L, prove that L, bisects BC.
In the given figure common tangents AB and CD to the two circles with centres O1 and O2 intersect at E. Prove that AB=CD
In the given figure, if ABC is an equilateral triangle. Find ∠BDC and ∠BEC.
In the given figure, chord EF || chord GH. Prove that, chord EG ≅ chord FH. Fill in the blanks and write the proof.
Is every chord of a circle also a diameter?
AB is a chord of a circle with centre O. AOC is diameter of circle, AT is a tangent at A.
Write answers of the following questions:
- Draw the figure using the given information.
- Find the measures of ∠CAT and ∠ABC with reasons.
- Whether ∠CAT and ∠ABC are congruent? Justify your answer.
