Tangents

A point on the circle, A point outside the circle

Step 1: Analyze each position relative to circle. Step 2: From point on circle: exactly one tangent exists. Step 3: From external point: exactly two tangents exist. Step 4: From internal point or center: no tangent possible as line would intersect circle.

Radius

Step 1: Consider the perpendicular from center to tangent. Step 2: This perpendicular distance from center to tangent line equals the radius. Step 3: This is because tangent touches circle at exactly one point, and this distance is the shortest path from center to tangent.

α = β, Both angles equal half the central angle, α - β = 0

Step 1: Apply tangent-chord angle theorem. Step 2: Angles made by tangents with chord on same side are equal (α = β). Step 3: Each angle equals half the central angle of the chord. Step 4: Since α = β, therefore α - β = 0.

12 cm

Step 1: Use Pythagoras theorem in right triangle formed. Step 2: Let T be point of contact, O be center. Step 3: In right triangle POT: PT² = PO² - OT². Step 4: PT² = 13² - 5² = 169 - 25 = 144. Step 5: PT = √144 = 12 cm.