Posted on April 22, 2022 (Updated on July 31, 2025)

How do you prove equal chords of a circle are equidistant from the Centre?

Space & Navigation

Cracking the Circle Code: Proving Equal Chords Keep Their Distance

Circles! Aren’t they just endlessly fascinating? All curves and symmetry, hiding secrets in plain sight. One of those secrets, a real gem in the world of geometry, involves something called chords. Specifically, it’s about how chords of equal length behave in relation to the circle’s center. The rule? Equal chords are always the same distance away from the center. Sounds simple, right? But proving it? That’s where the fun begins.

So, what exactly is a chord? Think of it as a straight line that connects any two points on the edge of the circle. Now, “equidistant” is just a fancy way of saying “the same distance.” In our case, we’re measuring the distance from the circle’s center to the chord itself.

Now, here’s the thing: it’s not just about knowing this rule. It’s about understanding why it’s true. That’s what really unlocks the power of geometry.

The Theorem and Its Flip Side

What we’re setting out to prove is this: If you’ve got two chords in a circle that are exactly the same length, then they’re sitting the same distance from the very center of that circle.

Interestingly, this rule works both ways. The flip side is also true: If you find two chords that are perfectly equidistant from the center, then guess what? They’re guaranteed to be the same length. It’s like a perfectly balanced equation.

Let’s Get to the Proof!

Alright, time to roll up our sleeves and get into the nitty-gritty. Here’s how we can prove that equal chords are indeed equidistant from the center:

  • Picture the Scene: Imagine a circle. Got it? Now, plop two chords inside, and make sure they’re exactly the same length. Let’s call them AB and CD.

  • Find the Middle Ground: Now, find the midpoint of each chord. Mark the midpoint of AB as M, and the midpoint of CD as N. So, AM is now equal to MB, and CN is equal to ND. Easy peasy.

  • Draw the Lines: Next, draw a straight line from the center of the circle (let’s call it O) to each of those midpoints (M and N). Make sure these lines are perfectly perpendicular (at a 90-degree angle) to the chords. These lines, OM and ON, are the distances we’re talking about – the distance from the center to each chord.

  • A Key Insight: Here’s a crucial piece of circle knowledge: a line drawn from the center of a circle that cuts a chord at a perfect right angle always slices the chord exactly in half. Because of this, AM = MB = AB/2 and CN = ND = CD/2. Since we already know that AB and CD are the same length, it means that AM and CN are also the same length.

  • Triangle Time: Now, let’s focus on the triangles OMB and OND. Look closely. OB and OD are both radii of the circle. That means they’re the same length!

  • Congruence Magic: Okay, time for some triangle magic. We’ve got:

    • ∠OMB = ∠OND (both are right angles, remember?)
    • OB = OD (they’re both radii)
    • MB = ND (we figured that out earlier)

    Because of all this, we can confidently say that triangle OMB is exactly the same as triangle OND. They’re congruent, thanks to the Right-angle Hypotenuse-Side (RHS) congruence rule.

  • The Grand Finale: Here’s the big payoff. Since triangles OMB and OND are carbon copies of each other, all their corresponding sides are equal. That means OM = ON! And that, my friends, is exactly what we wanted to prove.

    • The Big Picture

    So, there you have it. We’ve successfully proven that if two chords in a circle are the same length, they’re always going to be the same distance from the center. It’s a beautiful little piece of geometric logic that helps unlock even more of the circle’s secrets. Geometry, right? Who knew it could be this satisfying?

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