Tangents of circles problem (example 2) (video) | Khan Academy
circle diameter point of tangency arc center secant central angle minor arc .. If ZEOH GOF, what is the relationship between ⁀. EH and ⁀. FG?. Solve problems containing angles on or inside a circle. Another type of angle on a circle is one formed by a tangent and a chord. Explore, prove, and apply important properties of circles that have to do with things like arc length, radians, inscribed angles, and tangents.
Yes, depending on the chart they are filling out either the intercepted arc is double the inscribed angle, the angle whose vertex is in the interior of the circle is half the sum of the arcs, or the angle whose vertex is located outside the circle is half the difference of the two arcs.
How could you generalize this information? How could you explain that angle?
Angles in a Circle Theorems
The formulas listed above. The teacher could help the students arrive at the equations that the intercepted arc is double the inscribed angle, the angle whose vertex is in the interior of the circle is half the sum of the arcs, or the angle whose vertex is located outside the circle is half the difference of the two arcs.
How will the teacher introduce the lesson to the students? To begin the lesson students will participate in a "Win-Lose-Draw" game to review vocabulary words needed in the lesson.
How to play "Win-Lose-Draw": Write each of the vocabulary words above on index cards. So they must add up to degrees.
Intercepted arcs and angles of a circle (solutions, examples, videos)
So we get psi -- this psi plus that psi plus psi plus this angle, which is minus theta plus minus theta. These three angles must add up to degrees. They're the three angles of a triangle. Now we could subtract from both sides. Si plus psi is 2 psi minus theta is equal to 0. Add theta to both sides. You get 2 psi is equal to theta.
So we just proved what we set out to prove for the special case where our inscribed angle is defined, where one of the rays, if you want to view these lines as rays, where one of the rays that defines this inscribed angle is along the diameter. The diameter forms part of that ray. So this is a special case where one edge is sitting on the diameter. So already we could generalize this. So now that we know that if this is 50 that this is going to be degrees and likewise, right?
And now this will apply for any time. We could use this notion any time that -- so just using that result we just got, we can now generalize it a little bit, although this won't apply to all inscribed angles.
Inscribed angle theorem proof
Let's have an inscribed angle that looks like this. So this situation, the center, you can kind of view it as it's inside of the angle. That's my inscribed angle.
And I want to find a relationship between this inscribed angle and the central angle that's subtending to same arc. So that's my central angle subtending the same arc. Well, you might say, hey, gee, none of these ends or these chords that define this angle, neither of these are diameters, but what we can do is we can draw a diameter. If the center is within these two chords we can draw a diameter.
We can draw a diameter just like that.
If we draw a diameter just like that, if we define this angle as psi 1, that angle as psi 2. Clearly psi is the sum of those two angles. And we call this angle theta 1, and this angle theta 2. So si, which is psi 1 plus psi 2, so psi 1 plus psi 2 is going to be equal to these two things. Psi 1 plus psi 2, this is equal to the first inscribed angle that we want to deal with, just regular si.
What's theta 1 plus theta 2? Well that's just our original theta that we were dealing with. So now we've proved it for a slightly more general case where our center is inside of the two rays that define that angle.
Now, we still haven't addressed a slightly harder situation or a more general situation where if this is the center of our circle and I have an inscribed angle where the center isn't sitting inside of the two chords. Let me draw that. So that's going to be my vertex, and I'll switch colors, so let's say that is one of the chords that defines the angle, just like that.
Inscribed angle theorem proof (video) | Khan Academy
And let's say that is the other chord that defines the angle just like that. So how do we find the relationship between, let's call, this angle right here, let's call it psi 1. How do we find the relationship between psi 1 and the central angle that subtends this same arc? So when I talk about the same arc, that's that right there.
So the central angle that subtends the same arc will look like this. Let's call that theta 1. Let P be any other point onand join the interval OP. Hence P lies outside the circle, and not on it. This proves that the line is a tangent, because it meets the circle only at T. It also proves that every point onexcept for T, lies outside the circle. It remains to prove part b, that there is no other tangent to the circle at T.
Let t be a tangent at T, and suppose, by way of contradiction, that t were not perpendicular to OT.
Hence U also lies on the circle, contradicting the fact that t is a tangent. Tangents from an external point have equal length It is also a simple consequence of the radius-and-tangent theorem that the two tangents PT and PU have equal length. Tangents to a circle from an external point have equal length.
Tangents and trigonometry The right angle formed by a radius and tangent gives further opportunities for simple trigonometry.