Circle Secrets Revealed
The 5 Must-Know Rules of Chords and Angles
Mastering circle geometry hinges on understanding one thing: Chords. These five foundational theorems act like "cheat codes" that simplify complex geometry problems.
1. The Perpendicular Bisector Duo
This pair of theorems deals with the crucial relationship between the Center and the Chord.
Theorem 9.3 (Center to Chord)
The perpendicular drawn from the center to a chord bisects the chord.
Example: If line OM is ⊥ Chord AB, then AM = MB.
Theorem 9.4 (The Converse)
If you draw a line to the mid-point of a chord, it is perpendicular to that chord.
Meaning: It creates a 90° angle automatically.
2. Equal Chords & Central Angles
The length of a chord controls the angle at the center.
- Equal Chords = Equal Angles: If Chord AB = Chord CD, then ∠AOB = ∠COD.
- Equal Angles = Equal Chords: If the angles at the center are the same, the chords must be the same length.
💡 Simple Takeaway: The longer the chord, the wider the "V" shape at the center.
3. The Distance Rule
Remember, "distance" is always measured along the perpendicular line.
Equal Chords are Equidistant
If two chords are equal in length, they are the exact same distance from the center.
(Insight: The longest chord, the diameter, has a distance of zero!)
4. The "Double Angle" Rule (Most Important)
This is the theorem that appears most often in exams.
Theorem 9.7
The angle at the Center is double the angle at the Circumference.
Key Consequences:
- Angles in the Same Segment: All angles touching the circumference from the same arc are equal.
- Angle in a Semicircle: If the chord is a diameter, the angle at the edge is always 90°.
5. Cyclic Quadrilaterals
A quadrilateral is "Cyclic" if all 4 corners touch the circle.
| The Rule (Theorem 9.10) | Example |
|---|---|
| Opposite angles sum to 180°. | If ∠A = 100°, then opposite ∠C = 80°. |
🚀 Parent's Study Tip
These 5 rules act like mathematical levers. When my son solves circle problems, I tell him to look for the "Hidden Radius"—draw a line from the center to the edge. It often creates an isosceles triangle, making the problem easier to solve!
For more practice on this, visit our Sample Papers section.
Tip: Click the button above and select "Save as PDF" to practice offline.
