Home
Class 10
MATHS
In the given figure, O is the centre of ...

In the given figure, `O` is the centre of the circle and `TP` is the tangent to the circle from an external point `T`. If `/_PBT=30^(@)`, prove that `BA : AT =2 :1`.

Text Solution

Verified by Experts

`/_APB=90^(@)` [angle in a semicircle]
`:./_PAB=90^(@)-/_PBA=90^(@)-30^(@)=60^(@)` [`:' /_PBA=/_PBT=30^(@)`]
`/_PAT+/_PAB=180^(@)`[ linear pari]
`implies/_PAT=180^(@)-/_PAB=180^(@)-60^(@)=120^(@)`
`/_APT=/_PBA=30^(@)` [angles an alternate segments].
In `DeltaPAT`, we have
`/_APT+/_PAT+/_PTA=180^(@)`
`implies/_PTA=180^(@)-(/_APT+/_PAT)=180^(@)-(30^(@)+120^(@))=30^(@)`.
Now, `/_APT=/_PTA=30^(@)impliesAT=AP` ..........`(i)` [sides oppsite equal angles are equal]
In right `DeltaAPB`, we have
`cos(/_PAB)=(AP)/(BA)impliescos60^(@)=(AP)/(BA)impliesBA=2AP`. ..........`(ii)`
From `(i)` and `(ii)`, we get `BA : AT=2AP : AP=2 : 1`.
Promotional Banner

Topper's Solved these Questions

  • CIRCLES

    RS AGGARWAL|Exercise Exercise 8B|15 Videos

  • CIRCLES

    RS AGGARWAL|Exercise Multiple Choice Questions (Mcq)|15 Videos

  • CIRCLES

    RS AGGARWAL|Exercise Test Yourself|20 Videos

  • ARITHMETIC PROGRESSION

    RS AGGARWAL|Exercise Multiple Choice Questions (Mcq)|30 Videos

  • CONSTRUCTIONS

    RS AGGARWAL|Exercise Test Yourself|10 Videos

Similar Questions

Explore conceptually related problems

In the given figure, O is the centre of the circle, PA and PB are tangents to the circle then find angle AQB .

In figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If angleTPQ=70^(@) , find angleTRQ .

In the figure, O is the centre of a circle and AT is a tangent at point A. The measure of angle BAT is:

In given figure, O is the centre of the circle and LN is a diameter. If PQ is a tangent to the circle at K and angleKLN=30^(@) , find anglePKL .

In the figure, O is the centre of the circle and LN is a diameter. If PQ is a tangent to the circle at K and angle KLN = 30^(@) , find angle PKL .

In the given figure 'O' is the centre of the circle SP and TP are the two tangents at S and T respectively. angle SPT is 50^(@) , the value of angle SQT is:

In the figure, point P is the centre of the circle and line AB is the tangent to the circle at the point T. The radius of the circle is 6 cm . Find PB, if /_ TPB = 60^(@)