ICSE Class 10CISCE
Share
Notifications

View all notifications

Ab is the Diameter and Ac is a Chord of a Circle with Centre O Such that Angle Bac = 30°. the Tangent to the Circle at C Intersects Ab Produced in D. Show that Bc = Bd. - ICSE Class 10 - Mathematics

Login
Create free account


      Forgot password?
ConceptTangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments

Question

AB is the diameter and AC is a chord of a circle with centre O such that angle BAC = 30°. The tangent to the circle at C intersects AB produced in D. show that BC = BD.

Solution

 

Join OC,
`∠`BCD = `∠`BAC = 30° (angles in alternate segment)
Arc BC subtends `∠`DOC at the centre of the circle and `∠`BAC at the remaining part of the circle.
∴ `∠` BOC  = 2`∠`BAC =  2 × 30° = 60°
Now in Δ OCD,
`∠`BOC or `∠`DOC =  60°
`∠`OCD = 90° (OC ⊥ CD)
∴ `∠`DCO+ `∠`ODC =  90°
 ⇒ 60°  + `∠`ODC  = 90°
⇒ `∠`ODC  = 90° - 60° = 30°
Now in ΔBCD,
∵ `∠`ODC or `∠`BDC = `∠`BCD =  30°
∴ BC = BD

 

  Is there an error in this question or solution?

APPEARS IN

Solution Ab is the Diameter and Ac is a Chord of a Circle with Centre O Such that Angle Bac = 30°. the Tangent to the Circle at C Intersects Ab Produced in D. Show that Bc = Bd. Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.
S
View in app×