Similarity — Applications of Similarity in the Circle

About this lesson

This unit focuses on the relationships between chords, secants, and tangents in a circle, specifically when two lines containing chords or secants intersect either inside or outside the circle. It introduces the well-known problem that if two chords AB and CD of a circle intersect at point E, then EA × EB = EC × ED. It also covers the case where the intersection point is outside the circle, leading to the corollary that if EA is a tangent to the circle at A and EC is a secant intersecting the circle at D and C, then (EA)² = ED × EC. The unit includes worked examples and exercises where students find unknown lengths using these relationships, and it also presents the converse theorem: if EA × EB = EC × ED, then points A, B, C, and D lie on a circle. Additionally, it covers a corollary for proving that a segment is tangent to a circle if its square equals the product of the lengths of two secant segments. By the end of the unit, students should be able to apply these theorems to solve problems involving lengths in circles, prove that certain points are concyclic, and prove that a given segment is tangent to a circle.