High school geometry
- 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
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.
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- Can the trig function tan relate to a tangent of a circle? How?(12 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.(10 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.(7 votes)
- how can we find the radius of circle when c[h,k]= 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)
- 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)
- 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)
- cant you just do 3 squared minus 2 squared and you would get four(1 vote)
- dont you need to square root x because 4 is the square of x?(1 vote)
- 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.(1 vote)
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.