ICSE Class 10CISCE
Share
Notifications

View all notifications

In the Given Figure, Pat is Tangent to Circle with Centre O at Point a on Its Circumference and is Parallel to Chord Bc. If Cdq is a Line Segment (I) ∠Bap = ∠Adq (Ii) ∠Aob = 2∠Adq (Iii) ∠Adq = ∠Adb - ICSE Class 10 - Mathematics

Login
Create free account


      Forgot password?

Question

In the given figure, PAT is tangent to the circle with centre O at point A on its circumference and is parallel to chord BC. If CDQ is a line segment , show that:
(i) ∠BAP = ∠ADQ

(ii) ∠AOB = 2∠ADQ
(iii) ∠ADQ = ∠ADB

Solution

i) Since PAT ∥ BC

∴ `∠`PAB = `∠`ABC (alternate angles) .........(i)
In cyclic quadrilateral ABCD,
Ext `∠`ADQ = `∠`ABC ………..(ii)
From (i) and (ii)
`∠`PAB = `∠`ADQ
ii) Arc AB subtends AOB at the centre and ADB at the remaining part of the circle.
∴ `∠`AOB = 2`∠`ADB
⇒  `∠`AOB = 2`∠`PAB (angles in alternate segments)
⇒ `∠`AOB = 2`∠`ADQ (proved in (i) part)

iii)
∴  `∠`BAP = `∠`ADB (angles in alternate segments)
But
`∠`BAP = `∠`ADQ (proved in (i) part)
∴  `∠`ADQ = `∠`ADB

  Is there an error in this question or solution?

APPEARS IN

Solution In the Given Figure, Pat is Tangent to Circle with Centre O at Point a on Its Circumference and is Parallel to Chord Bc. If Cdq is a Line Segment (I) ∠Bap = ∠Adq (Ii) ∠Aob = 2∠Adq (Iii) ∠Adq = ∠Adb Concept: Cyclic Properties.
S
View in app×