Wednesday, January 27, 2010
Wednesday Math, Vol. 107: Inscribed Angles
You might well recall from a geometry class dimly shrouded in memory that the interior angles of a triangle add up to 180° and a full circle is 360°. That means the angles of the big triangle whose vertices are all on the circle add up to ½ the sum of the angles A + B + C in the picture to the left. The big triangle's angles are called inscribed angles, and the triangle itself is an inscribed triangle. The angles A, B and C are measured from the center of the circle and are therefore called central angles.
If we have a central angle whose rays touch the circle at two points and an inscribed angle whose rays also touch the circle at the same two points from the same side, the measure of the inscribed angle will be one half the measure of the central angle. In this picture, if we think of the letters as the measure of the angles, we get these three equations.
a + b = ½C
b + c = ½A
c + a = ½B
This is the result. Now comes the proof, if you desire to continue.
Because each of the smaller triangles in the picture have two radii as sides, they must be isosceles, and that means there are two angles of equal measure. This is shown in the picture by each small triangle having a capital letter measure at the central angle and two equal lowercase letters at the two inscribed angles. This gives us the following equations.
A + 2a = 180°
B + 2b = 180°
C + 2c = 180°
Add any two of these together. Let's choose the first two.
A + B + 2a + 2b = 360°
We also have the fact about the central angles.
A + B + C = 360°
2a + 2b = C.
Divide by two and we get
a + b = ½C.
The same can be done for the angles A and B by adding together different triangle pairs.
And as we say in Latin, Q.E.D., bitchez!