Grade 910 grade maths

O is the centre of the circle and TP is the tangent to the circle from an external point T. If angle PBT = 30 degree , prove that BA : AT = 2 : 1

Abrar Syed

10 Years agoGrade 9

2 Answers

Anwesha Barman

10 Years ago

Hi, I’m not able to upload the diagram ….anyway, here’s the solution.

Construction : Join P and O to get PO

Proof : Let Radius = r => BO=PO=AO=r

$\angle$ POA =2 $\angle$ PBA=2 x 30^o= 60^o [ $\angle$ at centre is twice the $\angle$ subtended on the circle.]

$\angle$ BPA = 90^o [ $\angle$ in semicircle.]

$\rightarrow$ $\angle$ PAB = 30^o[ by $\angle$ sum prop. in $\Delta$ PAB ]

In $\Delta$ POA,180 - ( $\angle$ O + $\angle$ A) = $\angle$ P => $\angle$ P =60^o

Therefore, $\Delta$ POA is euilateral.

Hence, PA=PO=AP =r

In $\Delta$ OPT, $\angle$ O =60^o , $\angle$ P = 90^o [tangent perpendicular to radius]

By $\angle$ sum prop., $\angle$ T = 30^o

On comparing, $\Delta$ BPA $\cong$ $\Delta$ TPO { above angles and PO=PA=r} byASA test

By CPCT, BA=TO

BO+OA=OA+TA

2r=r + TA

TA=r

BA/AT = 2r/r =2/1 = 2:1. …............proved.

Hope it helps....... :)

Yash Chourasiya

6 Years ago

Dear Student

AB is the chord passing through the center
So, AB is the diameter
Since, angle in a semicircle is a right angleRead more on Sarthaks.com -

I hope this answer will help you.
Thanks & Regards
Yash Chourasiya

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O is the centre of the circle and TP is the tangent to the circle from an external point T. If angle PBT = 30 degree , prove that BA : AT = 2 : 1

Abrar Syed

2 Answers

Anwesha Barman

Yash Chourasiya

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