Main content

## Geometry (all content)

### Unit 14: Lesson 9

Properties of tangents- Proof: Radius is perpendicular to tangent line
- Determining tangent lines: angles
- Determining tangent lines: lengths
- Proof: Segments tangent to circle from outside point are congruent
- Tangents of circles problem (example 1)
- Tangents of circles problem (example 2)
- Tangents of circles problem (example 3)
- Tangents of circles problems
- Challenge problems: radius & tangent
- Challenge problems: circumscribing shapes

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# Proof: Segments tangent to circle from outside point are congruent

CCSS.Math:

Sal proves that two tangent segments to a circle that are drawn from the same outside point are congruent.

## Want to join the conversation?

- I tried to make my own proof, is this valid? If you draw a line connecting each "tangent point" you will get another triangle, and now I had the 3rd angle so I put it as an equation 2x + 73 (that is the third angle) = 180, solved that which told me that the base angles are corresponding angles which means it must be an isosceles triangle, thus the lines are congruent.(17 votes)
- Actually, no. If you construct a triangle by drawing a line connecting the tangent points of the circle, the only way you could get that "2x" term in your equation is if you already assume that the triangle is isosceles (so that 2 of the 3 angles and 2 of the 3 sides would be congruent), which would directly imply the congruence of the tangent lines. To put it namely, your proof contains a fallacy: affirming the consequent, and it's therefore invalid.(10 votes)

- What property proves that the hypotenuses of the two right triangles are congruent?(2 votes)
- If they are the same line, you can use the Reflexive property to prove that they are congruent. If you can prove that the triangles are congruent, then the hypotenuses are also congruent. Also, the problem sometimes tells you that they are congruent.(2 votes)

- if the point o which is inside the circle is 117 degrees then would it double to figure out the measure of angle A?(2 votes)
- We are told that angles B and C are right angles, which add up to equal 180°. Since the angles in a quadrilateral add up to 360°, angle o plus angle A equal 360-180=180°. If, as you say, angle o is 117°, then angle A has to be 180-117=63°. Was that what you were asking?(2 votes)

- So just to check, a tangent line is a line that touches a circle at only one point right?(2 votes)
- Yes, a tangent line just touches a curve (in this video it is the curve of a circle) at only one point.(1 vote)

- Can I also use the RSH postulate instead of the Hypotenuse Leg Theorem?(1 vote)
- They are the same thing. HL, hypotenuse is only defined for a right triangle, so you need a right angle, the hypotenuse, and one of the legs.

RSH means right triangle, you need one of the sides and the hypotenuse.

The only difference is that one calls it a side and the other calls it a leg.(1 vote)

- Doesn't the Side Side Side Theorem prove that the hypotenuses of the 2 right triangle are congruent?(1 vote)
- Yes, it could, but there are more ways to prove that 2 hypotenuses are congruent.(1 vote)

- this dont make no sense(1 vote)
- wouldn't you be able to use ssa congruency instead of hypotenuse leg congruency ?

Please answer .

And thanks ahead of time(1 vote)- No SSA is not a proof of congruency. The reason HL works is that you have a hypotenuse and a leg, the Pythagorean theorem would always give you an equal third leg, so it is very related to SSS, not SSA. SSA always has two possibilities.(1 vote)

- I am working on tangents relating to circles and I have been stuck on this problem. It is a diagram showing three circles, all with different radii and who are touching one another. Then there is a triangle, not a right or iscoceles, that is connected between each of the circles' centers. The problem gives me the sides of the triangle and then wants me to find the radius of each circle.(1 vote)
- How can we know that any tangent has the same common point with the radius so that we know that it's perpendicular to it?(1 vote)
- The radius is a *line and has two edge points. One point starts from the center of a circle, the other end in a point of the edge of a circle. If you extend the radius, you would see that there is only a point that shares between the radius and a circle. The only way to draw a line that would touch the circle and shares the end edge of the radius on that point can draw only perpendicular to the radius.(1 vote)

## Video transcript

- [Voiceover] So I have a circle here, with a center at point O and
let's pick an arbitrary point that sits outside of the circle. So let me just pick this
point right over here. Point A. And if I have an arbitrary
point outside of the circle, I can actually draw two
different tangent lines that contain A, that are
tangent to this circle. Let me draw them. So one of them would look like this. Let me just start right over here. So I make a tangent to the circle. So it could look like that. And then the other one
would look like this, would look like, would look like that. Now let's say that the point that the tangent lines
intersect the circle, let's say this point right
over here, is point B, and this point right over here is point C. This is point C. Right over here. What I want to prove What I want to prove
is that the segment AB is congruent to the segment AC, or another way of thinking
about it, I want to prove, let me do this in a new color
that I haven't used yet, I want to prove that this
segment, right over here, is congruent to this segment... is congruent to that segment. I encourage you to pause the video and try to work it out on your own before I work through it. Alright, now let's try to
work through this together. And to work through this together, I'm going to actually
set up two triangles. Two triangles. And they're going to be right triangles as we'll see in a second. So let me draw some lines
here to set up our triangles. So I'm going to draw one line, just like that, and then we draw that. And then, let me draw that. Now what do we know about these triangles? Well, as I mentioned, we're going to be dealing
with right triangles. How do I know that? Well, in a previous video, we saw that if we have a radius
intersecting a tangent line, that they intersect at right angles. We've proven this. So that's a radius. That's a tangent line. They're going to intersect
at a right angle. So radius, tangent line, they
intersect at a right angle. We also know, since OB
and OC are both radii, that they're both the length
of the radius of the circle. So this side, right over here, let me do some new colors here, so this side is going to
be congruent to that side. And you can see that the
hypotenuse of both circles is the same side, side OA, so of course, it is equal to itself. This is equal to itself. So we see triangle ABO and triangle ACO, they're both right triangles
that have two sides in common, in particular, they both have a hypotenuse that are equal to each other and they both have a base, or a leg. And we know from Hypotenuse-Leg Congruence that if you have two right triangles, where the hypotenuses are equal, and you have a leg that are equal, then the both triangles
are going to be congruent. So triangle ABO is
congruent to triangle ACO. And in that proof where
we prove it, as well, the Pythagorean theorem tells us if you know two sides of a right triangle that determines what the third side is. So the third side, so length of AB is going to be the same
thing as the length of AC. Once again, it just comes out of, these are both right
triangles, if two sides, if two corresponding sides
of these two right triangles are congruent, then the third side has to, that comes straight from
the Pythagorean theorem. And there you go. We hopefully feel good about the fact that AB is going to be congruent to AC. Or another way to think about it, if I take a point outside of a circle, and I construct segments that
are tangent to the circle, that those two segments are going to be congruent to each other.