The proof is based on the geometric property that the lengths of tangents from a common external point to a circle are equal. Therefore, PA equals PB when both are tangents from point P to circle R.
The student's question concerns a typical proof in geometry related to tangents to a circle. When a line is tangent to a circle, it touches the circle at exactly one point. According to the properties of tangents to circles, the lengths of two tangents drawn from an external point to the circle are equal. This means that if we have point P outside a circle and from point P, we draw two tangents to the circle which touch the circle at points A and B, then PA will be equal to PB:
- Given: Lines from point P are tangent to circle R at points A and B.
- To Prove: PA = PB.
In a diagram, this would generally be displayed with a point P outside the circle, and two lines PA and PB touching the circle at A and B respectively. Because these two lines are tangents to the circle from the same external point, the distances from P to A and from P to B are equal by definition.
Therefore: The proof concludes that PA is equal to PB because the tangents from a common external point to a circle are always equal in length.