Proof of Tangent Secant Theorem
Consider the figure below, where O is the center of the circle ACD is secant of the circle and AB be the tangent on the circle. A line OP is drawn perpendicular to CD. Join OC, OA and OB.
Now, since OP β CD
CP = PD β(1)
[Perpendicular drawn from the center of the circle on the chord bisects the chord]
AC Γ AD = (AP β CP) (AP + PD)
β AC Γ AD = (AP β CP) (AP + CP) [From 1]
β AC Γ AD = AP2 β CP2
In β³ OAP
OA2 = OP2 + AP2
β AP2 = OA2 β OP2
β AC Γ AD = AP2 β CP2
β AC Γ AD = OA2 β OP2 β CP2
β AC Γ AD = OA2 β (OP2 + CP2)
In β³ OCP
OC2 = OP2 + CP2
β CP2 = OC2 β OP2
β AC Γ AD = OA2 β (OP2 + CP2)
β AC Γ AD = OA2 β (OP2 + OC2 β OP2)
β AC Γ AD = OA2 β OC2
Since OC = OB
Thus, AC Γ AD = OA2 β OB2
In β³ OAB
OA2 = OB2 + AB2
β AB2 = OA2 β OB2
β AC Γ AD = OA2 β OB2
β AC Γ AD = AB2
Hence proved
Tangent Secant Theorem
Tangent Secant Theorem is the fundamental theorem in geometry. Tangent and secant are the important parts of the circle. The tangent secant theorem is used in various fields of mathematics, construction, and many more. Tangents and secants are the lines that intersect the circle at some points.
In this article, we will learn about the Tangent Secant theorem in detail along with its statement and proof. It also covers the applications and limitations of the tangent secant theorem and some solved examples of the Tangent Secant Theorem. Letβs start our learning on the topic Tangent Secant theorem.
Table of Content
- What is Tangent and Secant?
- What is Tangent Secant Theorem?
- Proof of Tangent Secant Theorem
- Limitation and Applications of Tangent Secant Theorem
- Solved Problems
- FAQs
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