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# Tangents of circles problem (example 3)

Video transcript

Line AC is tangent to
circle O at point C. So this is line AC, tangent
to circle O at point C. What is the
length of segment AC? What is this distance right over
here, between point A and point C? And I encourage you
now to pause this video and try this out on your own. So I'm assuming you've
given a go at it. So the key thing
to realize here, since AC is tangent to the
circle at point C, that means it's going to be
perpendicular to the radius between the center of
the circle and point C. So this right over
here is a right angle. And the reason
why that is useful is now we know that triangle
AOC is a right triangle. So if we know two
of its sides, we could use the
Pythagorean theorem to figure out the third. Now, we clearly know OC. Now OA, we don't
know the entire side. They only give us
that AB is equal to 2. But the thing that might
jump out in your mind is OB is a radius. It's going to be the same
length as any radius. So this is going
to be 3 as well. It's the distance between
the center of the circle and a point on the circle, just
like the distance between O and C. So this is
going to be 3 as well. And so now we are
able to figure out that the hypotenuse of
this triangle has length 5. And so we need to figure out
what the length of segment AC is. So let's just call
that, I don't know. I'll call that x. And so we know that x
squared plus 3 squared-- I'm just applying the
Pythagorean theorem here-- is going to be equal to the
length of the hypotenuse squared, is going to
be equal to 5 squared. And I know this
is the hypotenuse. It's the side opposite
the 90-degree angle. It's the longest side
of the right triangle. So x squared plus
9 is equal to 25. Subtract 9 from
both sides, and you get x squared is equal to 16. And so it should jump
out at you that x is going to be equal to 4. So x is equal to 4. x is the same thing as
the length of segment AC, so the length of
segment AC is 4.