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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?
• Wait a second, couldn't Mr. Sal use the pythagorean triple 3, 4, 5. I'm just curious why didn't he use it.
• 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.
• how can we find the radius of circle when c[h,k]=[00] and tangent to the line ix=-5 ?
• 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.
• In the problem x^2+12^2=x^2+16x+64, where do you get the 16?
• dont you need to square root x because 4 is the square of x?
• 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.
• How would I find the length of a quadrilateral formed from two tangent at a circle when only the radius is given?
• 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.
• 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.