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Secants - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

A secant is a line, ray, or line segment that intersects a circle in two places. Three points are covered: (1) secants that intersect in a circle which divide each other proportionally, (2) the angle formed by secants which intersects in a circle and is half the sum of the intercepted arcs, and (3) two secants drawn from the same point outside a circle that form an angle whose measure is half the difference of the intercepted arcs.

Secants are lines that intercept the circle in two places, so notice that I could erase one of these arrows creating a ray and this would also be considered a secant because it starts on the outside and it passes through the circle. Let's say I had a line segment like this that would also be considered a secant because it is intersected the arc, excuse me it's intersect the circle in two places, so couple of key things of note about secants.
The first is when you have an intersection of 2 secants that's inside the circle. You're going to create 2 different angles, we call this angle 1 and angle 2 where we know that those are going to be supplementary since they're linear pair. What's interesting is that if you added up the intercepted arcs of angle 1 so if I call this arc x and if I call the other intercepted arc y, if I added these up and divided by 2 that would be the measure of angle 1 so I wrote that down here in our equation measure of angle 1 is equal to half the sum of the intercepted arcs.
Second key thing about secants is that when they intersect each other, they divide each other proportionally so way to write that Mathematically is to say that the length of a times the length of b is equal to the length of x the length of y, so the 2 pieces of one secant when you multiply them is going to equal the 2 segments of the other segment when you divide them.
Third key thing is when we have a point outside the circle and we draw two secants, notice that these are not tangent. What is interesting is that this angle right here that is formed by these 2 rays is equal to half of the difference, not the sum but the difference, of the larger arc minus the smaller arc, so these 3 key things are going to help you solve for missing angles in problems.

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