lesson
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Imagine drawing a straight line across a paper plate, then folding the plate right through the middle. If you fold it at a sharp square corner, that line splits into two matching halves.
A straight line connecting any two points on a circle's edge is called a chord, while a line that cuts another segment exactly in half is said to bisect it.
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What happens every single time a line from the center drops down at a right angle to any chord?
The Perpendicular Bisector Rule
When a line from the center of a circle meets a chord at a 90-degree angle (perpendicular), it cuts the chord into two equal lengths.
Over 2,300 years ago, the Greek mathematician Euclid wrote this exact relationship down in his famous geometry guide called Elements.
Why must these two halves always be perfectly equal? Let's look at the secret triangles hidden inside.
Why It Works: The Triangle Proof
Draw straight lines from center O out to points A and B. These lines are radii, and because every radius in a circle has the same length, OA=OB.
Now look at the two right-angled triangles side by side: △OMA and △OMB. They share the middle side OM, have matching right angles, and have identical hypotenuse lengths OA=OB.
📊Interactive diagram
Because both triangles are identical twin shapes (congruent), their bottom edges must also match: AM=MB.