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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.

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  • primosaur tree style avatar for user √ǺɌǏȚȒǍɸ
    Can the trig function tan relate to a tangent of a circle? How?
    (12 votes)
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  • hopper cool style avatar for user Ohm Rajpal
    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)
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    • mr pink green style avatar for user David Severin
      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)
  • blobby green style avatar for user islamkot100
    how can we find the radius of circle when c[h,k]=[00] and tangent to the line ix=-5 ?
    (2 votes)
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    • mr pink red style avatar for user andrewp18
      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)
  • starky tree style avatar for user Devon Fodrie
    In the problem x^2+12^2=x^2+16x+64, where do you get the 16?
    (2 votes)
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  • blobby green style avatar for user Mcmurtry1900
    How would I find the length of a quadrilateral formed from two tangent at a circle when only the radius is given?
    (2 votes)
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  • duskpin ultimate style avatar for user ★彡 ✦StarLight✦ 彡★
    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)
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  • starky sapling style avatar for user Hodorious
    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)
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    • piceratops ultimate style avatar for user Fai
      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)
  • blobby green style avatar for user Gregory  Gentry
    Sal is always applying the Pythagorean Theorem to everything WHY?
    (1 vote)
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  • primosaur seed style avatar for user isy
    cant you just do 3 squared minus 2 squared and you would get four
    (1 vote)
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  • aqualine ultimate style avatar for user 1.queen.elisabeth
    dont you need to square root x because 4 is the square of x?
    (1 vote)
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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.