Circles: Chords, Secants and Tangent — Part 2: Product of segments and the alternate segment theorem: visual lesson, ICSE Class 10 Mathematics

Lesson

Circles: chords and tangents, Part 2

When two chords cross inside a circle, the products of their parts are equal. In this part we prove that, extend it to secants and tangents, and finish with the alternate segment theorem.

In this lesson you will

Prove and use the intersecting chords theorem
Use the same product rule for secants that meet outside the circle
Prove and use the tangent-secant theorem
Prove and use the alternate segment theorem to find angles

Chords crossing at $P$: $6 \times 10 = 60 = 12 \times 5$.

Concept

What you will be able to do

Prove and use the intersecting chords theorem
Use the same rule for secants from an outside point
Prove and use tangent squared equals product of a secant
Prove and use the alternate segment theorem

Part 2: chords, secants and the alternate segment.
Theorem	In words	Figure
Intersecting chords	PA times PB equals PC times PD	two chords crossing inside
Secants from a point	PA times PB equals PC times PD	two secants from outside
Tangent and secant	PT squared equals PA times PB	one tangent and one secant
Alternate segment	tangent-chord angle equals the angle in the alternate segment	a tangent and chords from the point of contact

Concept

Intersecting chords

If two chords $AB$ and $CD$ of a circle cut at a point $P$ inside the circle, then the product of the parts of one chord equals the product of the parts of the other.

here $5 \times 12 = 60$ and $6 \times 10 = 60$

PA \times PB = PC \times PD

Intersecting chords: $PA \times PB = PC \times PD$, here $5 \times 12 = 6 \times 10$.

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