About this video
The video proves that if a diameter bisects a chord, it is perpendicular to the chord. Drawing two radii makes two triangles that are congruent by side-side-side, so the angles at the chord are equal; they are also supplementary, so each is 90 degrees.
You'll be able to
- explain what it means for a diameter to bisect a chord.
- prove two triangles congruent by SSS using equal radii and a shared side.
- show that two equal supplementary angles must each be 90 degrees.
Key words
- bisect
- To split something into two equal parts.
- diameter
- A chord that passes through the centre of the circle.
- perpendicular
- Meeting at a right angle (90 degrees).
- side-side-side (SSS)
- If all three sides of one triangle equal the three sides of another, the triangles are congruent.
- supplementary
- Two angles that add up to 180 degrees.
Transcript
- [Instructor] In this video, we wanna prove that if a diameter of a circle bisects a chord, it is perpendicular to the chord. So we have drawn the first part, we have a diameter here, and it is bisecting this chord. And bisecting just means it splits it into two equal segments.
So that's why we have the lines there. That's what the bisects tell us. And, now, how do we determine whether it's truly perpendicular? Well, what I'm gonna do is construct some triangles. So let me put some radii out here. So that's one radius, and then that's another radius.
This is a circle, so the length of the radius is always the same. So that length is equal to that length. And then we also know, of course, this side is equal to itself. So all of a sudden we have these two triangles that are congruent to each other. How do we know?
Well, it's by side, side, side. They both share this side, they both have a side that is these double slashes, and they both have this side that's on that bisected chord. So we know that these two triangles are congruent. Well, that means that corresponding angles are also going to be congruent.
And let me do this in a new color. This angle, let's call that theta. The measure of this angle is theta. And let's call the measure of this angle phi. We know that these measures have to be equal to each other, that theta needs to be equal to phi because they're corresponding angles of two congruent triangles.
But we also know, because they're colinear, that they're supplementary. We also know that theta plus phi is going to be equal to 180. Well, if theta equals phi, we could just substitute phi for theta here. And then we would get 2 theta is equal to 180. Divide both sides by 2, and you get theta is equal to 90.
So theta is 90, and if phi is also equal to theta, they all equal 90. And so the diameter in question and the chord that it bisects, they are perpendicular. We have just shown that.
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