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Pt is a Tangent to the Circle at T. If ∠Abc = 70° and ∠Acb = 50°; Calculate: (I) `∠`Cbt (Ii) `∠`Bat (Iii) `∠`Apt - ICSE Class 10 - Mathematics

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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

PT is a tangent to the circle at T. if ∠ABC = 70° and ∠ACB = 50°; Calculate:
(i) `∠`CBT (ii) `∠`BAT (iii) `∠`APT  

Solution 1

 

Join AT and BT.
i) TC is the diameter of the circle
∴ `∠`CBT =  90°  (Angle in a semi – circle)
(ii) `∠` CBA =  70°
∴ `∠`ABT = `∠`CBT - `∠`CBA = 90°  - 70 °  = 20°
Now, `∠` ACT  = `∠`ABT =  20°  (Angle in the same segment of the circle)

∴`∠` TCB = `∠`ACB - `∠`ACT =  50° - 20° = 30°
But, `∠` TCB  = `∠`TAB (Angles in the same segment of the circle)
∴ `∠` TAB or `∠` BAT = 30° 
(iii) `∠`BTX  = `∠`TCB = 30 °(Angles in the same segment)
∴ `∠` PTB = 180° - 30° =150°

Now in Δ PTB
`∠` APT  + `∠`PTB  +`∠` ABT = 180°
⇒ `∠`APT + 150 ° + 20° =  180°
⇒ `∠` APT =  180° -  170° = 10°

 

Solution 2

 

Join AT and BT.
i) TC is the diameter of the circle
∴ `∠`CBT =  90°  (Angle in a semi – circle)
(ii) `∠` CBA =  70°
∴ `∠`ABT = `∠`CBT - `∠`CBA = 90°  - 70 °  = 20°
Now, `∠` ACT  = `∠`ABT =  20°  (Angle in the same segment of the circle)

∴`∠` TCB = `∠`ACB - `∠`ACT =  50° - 20° = 30°
But, `∠` TCB  = `∠`TAB (Angles in the same segment of the circle)
∴ `∠` TAB or `∠` BAT = 30° 
(iii) `∠`BTX  = `∠`TCB = 30 °(Angles in the same segment)
∴ `∠` PTB = 180° - 30° =150°

Now in Δ PTB
`∠` APT  + `∠`PTB  +`∠` ABT = 180°
⇒ `∠`APT + 150 ° + 20° =  180°
⇒ `∠` APT =  180° -  170° = 10°

 

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Solution Pt is a Tangent to the Circle at T. If ∠Abc = 70° and ∠Acb = 50°; Calculate: (I) `∠`Cbt (Ii) `∠`Bat (Iii) `∠`Apt 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.
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