Tangents of circles problem (example 3)
About this video
This video finds the length of tangent segment AC. Because a tangent is perpendicular to the radius at the point of tangency, triangle AOC is a right triangle. The radius is 3, and OA = OB + AB = 3 + 2 = 5, so the Pythagorean theorem gives AC = 4.
You'll be able to
- recognize the right triangle formed by a radius, a tangent segment and the segment to the center.
- use the fact that all radii of a circle are equal to find missing lengths.
- use the Pythagorean theorem to find a tangent segment length.
Key words
- tangent
- A line that touches a circle at exactly one point and is perpendicular to the radius there.
- radius
- The distance from the center of a circle to any point on it; all radii are equal.
- hypotenuse
- The side opposite the right angle; the longest side of a right triangle.
- Pythagorean theorem
- In a right triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2.
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.
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