Key relationships between chords and the center — how perpendiculars, bisection, and distance all connect.
The Mnemonic
Two Linked Facts About Chords and the Center
Theorem 1: A perpendicular segment drawn from the circle's CENTER to a chord always BISECTS that chord (cuts it into two equal halves). This also works in reverse: a segment from the center to a chord's midpoint is always perpendicular to that chord.
Theorem 2: Two chords are CONGRUENT (equal length) if and only if they are EQUIDISTANT from the center (the same perpendicular distance away). Longer chords are always closer to the center; shorter chords are always farther away.
The perpendicular from the center bisects a chord — and the shorter chord (blue) sits farther from center only if it's actually shorter; equal chords sit equidistant from center
💡 Memory Trick
"Key relationships between chords and the center." Perpendicular from center → bisects the chord, always. Equal chords → equal distance from center, always (and this works both directions — equal distance guarantees equal chords too). Distance and length trade off: closer to center means longer chord.
Why It Works
Both Theorems Come From the Same Isosceles Triangle
Draw two radii to the endpoints of a chord — since both are radii, they're equal, making an isosceles triangle with the chord as its base. The perpendicular from the center to the chord is exactly the ALTITUDE to that base. Recall from Triangle Line Segments that in an isosceles triangle, the altitude from the apex is always the SAME segment as the median — and a median, by definition, goes to the midpoint. That's exactly why the perpendicular from the center bisects the chord.
For the second theorem: since every chord forms this same kind of isosceles triangle with two radii (of the SAME fixed length, since all radii in one circle are equal), two chords will only produce triangles with the same perpendicular height (distance from center) when the chords themselves are the same length — this connects directly to the Pythagorean Theorem, since (half-chord)² + (distance from center)² = radius² for any chord, and the radius is fixed.
Using It In A Proof
Using the Right Triangle These Theorems Create
Since the perpendicular-bisector relationship creates a right triangle (radius as hypotenuse, half-chord and distance-from-center as legs), most problems reduce to the Pythagorean Theorem once that triangle is identified.
1
Draw the perpendicular from the center to the chord
This creates a right angle and bisects the chord into two equal halves.
2
Identify the right triangle
The radius is the hypotenuse; half the chord's length and the distance from the center are the two legs.
3
Apply the Pythagorean Theorem
(half-chord)² + (distance from center)² = radius² — solve for whichever value is missing.
Full Worked Example
Finding a Chord's Length From Its Distance to the Center
Given: A circle has radius 13. A chord is 5 units from the center. Find: the chord's full length.
1
Set up the right triangle
Radius (hypotenuse) = 13. Distance from center (one leg) = 5. Half the chord (other leg) = unknown.
Notice 5-12-13 is a Pythagorean triple — recognizing this pattern (from Pythagorean Theorem) made the arithmetic immediate rather than requiring a square root calculation.
🎯 Quick Worked Example
Two chords in the same circle are both 16 units long. If one chord is 6 units from the center, how far is the other chord from the center?
1
Apply the equal-chords theorem. Since both chords are the same length (16), they must be equidistant from the center.
2
Apply the given distance. The first chord is 6 units from the center.
3
Conclude. The second chord must ALSO be exactly 6 units from the center, since equal chords are always equidistant.
📌 Exam Application
The equal-chords-equal-distance relationship is frequently tested WITHOUT any calculation at all — simply recognizing that two chords with the same given length must be the same distance from the center (or vice versa) is often the entire question, no Pythagorean Theorem needed.
⚠️ Most Common Chord-Distance Theorems Mistakes
Trap 1 — Using the full chord length instead of half in the Pythagorean setup: The perpendicular from the center bisects the chord, so it's HALF the chord length that forms the leg of the right triangle — using the full chord length instead gives a wrong equation.
Trap 2 — Assuming longer distance means longer chord: The relationship is actually the OPPOSITE — a chord closer to the center is LONGER, and a chord farther from the center is SHORTER (the diameter, at distance 0, is the longest possible chord).
✓ Quick Self-Test
1) What does a perpendicular from the center to a chord always do to that chord? 2) What is true about two chords that are equidistant from the center? 3) A circle has radius 10, and a chord is 6 units from the center — find the chord's length. 4) Which is longer: a chord 2 units from the center, or a chord 7 units from the center? 5) Why does the perpendicular-bisector relationship connect to the Isosceles Triangle Theorem?