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Edexcel IGCSE Maths 復(fù)習(xí)筆記 4.7.1 Intersecting Chord Theorem

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Edexcel IGCSE Maths 復(fù)習(xí)筆記 4.7.1 Intersecting Chord Theorem

What are circle theorems?

Circle?theorems?are?angle?(and?distance)?properties?that arise when lines and shapes drawn within or connected to a circle
There are quite a few of them ...
- Circle Theorems – Angles at Centre & Circumference
- Circle Theorems – Tangents
- Circle Theorems – Cyclic Quadrilaterals
- Circle Theorems – Alternate Segment
This set of notes deals with the?Intersecting Chord Theorem
But first make sure you are familiar with the?names?of?parts?of a circle

4.7.1-InChTh-Notes-fig1

What is the intersecting chord theorem?

4.7.1-InChTh-Notes-fig2

For two chords,?AB?and?CD?that meet at point P
- AP : PD?≡?CP : PB
- Ratio?of?longer?lengths (of chords) ≡?Ratio?of?shorter?lengths (of chords)
An more practical way to deal with most problems is
- AP ×PB?= CP × PD
You do not need to know the proof this theorem
- You may be able to see a loose connection to?similar?shapes

How do I use the intersecting chord theorem to solve problems?

4.7.1-InChTh-Notes-fig3

Keep track carefully of which distance is associated with each part of each chord

Harder problems with intersecting chord theorem

4.7.1-InChth-Notes-fig4

The algebra can be made harder by having more, and more awkward expressions for the distances involved

Exam Tip

If you do not like the capital letter notation used you can rename the lengths of the chord using single letters (see the second diagram above).The multiplication version of the theorem is easier to remember and work with but you may be asked questions about ratios too (see the Worked Example, part (b), below).