Inscribed Angle Theorems
Part 1: What is an Inscribed Angle?
An angle made from points sitting on the circle's circumference.
Move "D" and Observe the "inscribed angle".
Q : How about the angle when you move the point "D"? Did you see any change of the angle?
Move "C" and Observe the "inscribed angle".
What factors decide the inscribed angle?
Which factors change the angle and which factors do not change the angle? Compare 1 and 2.
Part 2: Angle at the Center Theorem
As we saw in the part 1, an inscribed angle is always the SAME as long as the opposite side from the point is fixed on a circle. This is the first part of the "Inscribed Angle Theorems".
Next, let's see the second part, an angle at the center!!
Move "E" between "D" and "A" and Observe the change of an angle.
Q: Find the ratio
You can find a very simple ratio between BDC and BAC.
Using exterior angle is one way...
Though this is not a complete proof, it will help your understanding...
BDC=a +b
BAC=a+a+b+b
BAC=2BDC
An inscribed angle is Half of the central angle!!
A central angle is Double of an inscribed angle!!
Move "C" and "I" . The opposite side is on a "CIRCLE'S DIAMETER" !!
It is called "Thales' Theorem"
As you see, it is always the right angle.
You can use Inscribed Angle Theorems here too.
The center angle is 180 so the inscribed angle is ....half size..... yes, 90!!!,
Self review points!
1 Inscribed angle.
2 Inscribed angle theorem part 1 about the same angle on a fixed length opposite side.
3 Inscribed angle theorem part 2 about 1:2 ratio between the inscribed angle and the center angle.
4 Special case of the inscribed angle theorem, Thales' Theorem.
See you!