Answer:
Step-by-step explanation:
Given: Circle O with BT as a tangent at point B.
T is the midpoint of arc CD.
To prove: (CA)(TB) = (TA)(CT)
Since point T is the midpoint of arc CD,
Therefore, m(arc CT) = m(arc TD)
and m∠CAT ≅ m∠DAT
m(∠ACT) = 90° [Angle subtended by the diameter]
By applying tangent rule in triangles ACT and ABT,
tan(∠CAT) = [tex]\frac{CT}{CA}[/tex]
Similarly, tan(∠BAT) = [tex]\frac{TB}{TA}[/tex]
Since tan(∠CAT) = tan(∠BAT)
Therefore, [tex]\frac{CT}{CA}=\frac{TB}{TA}[/tex]
(CA) × (TB) = (CT) × (TA)
Hence proved.
