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### Course: High school geometry > Unit 8

Lesson 10: 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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# Tangents of circles problem (example 3)

Sal finds a missing length using the property that tangents are perpendicular to the radius. Created by Sal Khan.

## Want to join the conversation?

- Can the trig function tan relate to a tangent of a circle? How?(14 votes)
- Yes. The tangent line corresponds to one of the sides of a triangle that is tangential to the point
`(cosθ, sinθ)`

. I can't find a great article specifically on tangent, but this picture shows the tangent line: http://upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle-trig6.svg/1000px-Circle-trig6.svg.png Note that you can use similar triangles to show that the line in brown is equal to`sinθ / cosθ`

.(12 votes)

- Wait a second, couldn't Mr. Sal use the pythagorean triple 3, 4, 5. I'm just curious why didn't he use it.(4 votes)
- You are correct, but the purpose of the video might help when the numbers are not that simple. The hardest one would be trying to find the radius given other information. While you know the answer to the specific question quickly, it would not help on the process of solving similar prolblems.(8 votes)

- how can we find the radius of circle when c[h,k]=[00] and tangent to the line ix=-5 ?(2 votes)
- There is a lovely formula:

|𝑎𝑥₁ + 𝑏𝑦₁ + 𝑐|/√(𝑎² + 𝑏²)

This formula tells us the shortest distance between a point (𝑥₁, 𝑦₁) and a line 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0. Since the radius is perpendicular to the tangent, the shortest distance between the center and the tangent will be the radius of the circle.

𝑥 = 5

This can be rewritten as:

𝑥 - 5 = 0

Fitting this into the form:

𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0

We see that:

𝑎 = 1

𝑏 = 0

𝑐 = -5

Now the center of the circle (𝑥₁, 𝑦₁) is simply (0, 0). Plugging this all into the formula gives us:

𝑟 = 5

Now I gave you a very long explanation but with intuition, you should've been able to realize that, centered at the origin and ending at 𝑥 = 5, the circle must have had a radius of 5. The formula will help with more confusing centers and tangents.

Comment if you have questions.(4 votes)

- In the problem x^2+12^2=x^2+16x+64, where do you get the 16?(2 votes)
- dont you need to square root x because 4 is the square of x?(2 votes)
- well, using the pythagorean theorem, you have a^2+b^2=c^2. when you have x^2=16, you need to square root both x^2 and 16, so you can find out the value of x. in this case, x=4.(2 votes)

- How would I find the length of a quadrilateral formed from two tangent at a circle when only the radius is given?(2 votes)
- Okay . . . but how do you do it with only the length of the radius and two angles? This was in a test yesterday and my teacher said something about trig ratios, which I FRANKLY did not get. Here Sal has the lengths of the hypotenuse and the radius (the opposite side), but I only had the radius . . . and two angles.

Anyone who can clear this up for me? Thanks!(1 vote)- Assuming the two angles were in a right triangle, you would use sine, cosine, and or tangent using the angles and the radius to find the other missing side length(s).

Use SOH CAH TOA for the correct ratios.(2 votes)

- When we say that a certain line is tangent to circle O, do we assume that O is the center of the circle? And when referring to circles in general, is it enough to use one point or do we need to refer to at least two?(1 vote)
- O would be the center of the circle. Usually circles are defined by two parameters: their center and their radius. Usually referring to a circle by only one parameter is only valid when you are solving a geometry problem where a diagram is provided and clearly labelled.(2 votes)

- and you could easily figure out x if you knew your pythagorean triples because if a leg is 3 and the hypotenuse is the larger number 5 (same with side of 4 hypotenuse cant be 3 because hypotenuse is larger than a leg) you can already know that x is 4. Pythagorean triples have made my life SO much easier(2 votes)
- I agree! But this really should go under Tips and Thanks, so I flagged you for it. No offense. =)(0 votes)

- Sal is always applying the Pythagorean Theorem to everything WHY?(1 vote)
- The reason Sal applies the Pythagorean theorem so often is that it is the simplest way to find side lengths-a special form of the sine rule.(2 votes)

## 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.