Proof: The product of segments of intersecting secant and tangent
About this video
The video proves that for a secant segment AC (with external part AB) and a tangent segment AD from the same outside point, AC x AB = AD^2. It shows triangles ABD and ADC are similar by angle-angle, using an inscribed angle and a tangent-chord angle that intercept the same arc, plus the shared angle A, then cross-multiplies the side ratio.
You'll be able to
- state the secant-tangent rule: whole secant x external part = tangent squared.
- explain why triangles ABD and ADC are similar.
- use AC x AB = AD^2 to find a missing length.
Key words
- secant segment
- A segment from an outside point that passes through the circle, crossing it twice.
- tangent segment
- A segment from an outside point to the single point where it touches the circle.
- external part
- The piece of a secant segment between the outside point and the circle.
- tangent-chord angle
- The angle between a tangent and a chord; it is half the intercepted arc.
- similar triangles
- Triangles with equal angles, so corresponding sides are in proportion.
Transcript
- [Instructor] What we wanna prove in this video is that if we take the product of the length of a secant segment, so that was AC, and the part of that segment that is external to the circle, so AC times AB, that that's actually going to be equal to the square of the length of a tangent that sits outside of the circle.
Well, that's always gonna happen for a tangent. But the length of the tangent that intersects, that intersects that secant segment. So it's gonna be equal to the square of segment AD. That's what we're gonna prove here. And as you can imagine, what we're gonna do is try to set up some similar triangles and set up some proportional relationship and then do a little bit of math and hopefully we can get this result.
Now, what we're going to do is set up a, set up two things. So we could set up a triangle like this, and then we can also set up a triangle like that. Now, why is that useful? Well, if we were to say that, well, I don't even have to name what the measure is of this arc over here, but we know whatever that measure is of that arc, this angle over here is going to be half of that, because it's an inscribed angle that intersects that arc.
And then this angle right over here, BDA, that's an angle formed between a tangent and a cord. And that also where we know that that arc that it intersects that this angle's going to be half that. So this is going to be half that arc measure, and this is going to be half that arc measure.
We know that these are going to have the same measure, and now we also know that, well, if this angle up here is in two different triangles, it's going to be, I guess you could say congruent to itself. So let's put it there. And so now we can see a relationship.
If we think about triangle ABD, let me write this down. So let's think about the relationship triangle ABD, and that's this triangle here. There's triangle, ABD. Let's think about the relationship between that and triangle, and I'm gonna try to get the same and triangle ADC.
Triangle ADC. A, D, and C. They have two angles in common. Both of those triangles have both this red one and this off-white colored one. So by angle angle they are going to be similar, similar by angle angle. If you have two angles in common, you're gonna actually have the three angle, third angle as well.
So now this allows us to set up a proportion. We could say, we could say that the ratio of, let's look at the big triangle first, the ratio of the length of AC, the longest side, to its shortest side, AD, is going to be equal to the ratio. Let's look at the smaller triangle.
Now, the ratio of its longest side, AD, that goes between the red angles and the white angle. The way I've colored it right over here. It's longest side and it's shortest side, at least the way it's been drawn. You could also say the side between the white angle and the one that is not labeled so far with a color or angle marks.
Angle measure marks, so AD to AB. Now let's multiply both sides by both denominators. We're gonna multiply this times AB and AD, and we're gonna multiply this side by AB and AD. And what do we get? This cancels with this. On the left hand side we have AC times AB.
And on the right hand side, this cancels with that. We have AD squared. AD squared, and we're done.
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