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Current time:0:00Total duration:3:56

Proof: Segments tangent to circle from outside point are congruent


Video transcript

so I have a circle here with the 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 actually let me just start right over here so let make it 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 and let's say that the point that the tangent lines intersect the circle let's say at this point right over here is point B and this point right over here is point this is point C right over here what I want to prove what I want to prove is that the segment a B 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'd encourage you to pause to pause the video and try to work it out and try to work it out on your own before I work through it all right now let's try to work through this together into it 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 let me draw some lines here to set up our triangles so I'm going to draw one line and just like that and then let me draw that and then let me draw that now what do we know 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 previous video we saw that if we have a radius intersecting a tangent line that the that they intersect at right angles we prove we've proven this that's a radius that's a tangent line they're gonna intersect at a right angle it's a 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 o a so of course it is equal to itself this is equal to itself and so we see triangle a B oh and triangle AC oh 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 hypotenuse is are equal and you have a leg that are equal then the two triangles are going to be congruent so triangle [ __ ] is congruent to triangle a Co and in that proof where we prove it as well if you 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 the length of a B is going to be the same thing as the length of AC and that's once again 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 two that comes straight from the Pythagorean theorem and there you go we've hopefully feel good about the fact that a B is going to be congruent to AC or another way to think about 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