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1. Angles and Lines
2. Intersecting Angles
3. Triangles
4. Circles
5. Perimeters & Areas
6. Solids
7. Coordinate Geometry

Circles
   The diameter, d, of a circle is twice the radius, r. Its circumference is d or 2r ( = 3.14 or 22/7- which is approximately 3.14). A central angle has its vertex at the center of a circle, and its measure equals the measure of the arc it intercepts (in degrees). For example, if AOB = 60,

Circumference = 2r =d

AOB = arc AB

 

then the measure of arc AB is 60, or 60/360 = 1/6 of the circle's circumference. An inscribed angle has its vertex on the circle itself, and its measure is 1/2 of the measure of the arc it intercepts:

ACB = 1/2 arcAB.

A line that just touches a circle is called a tangent. It is perpendicular to the radius drawn to the point of touching.

ABC is a right triangle if CB is the diameter. A triangle inscribed in a circle is a right triangle if one of its sides is a diameter. Obviously, A has its vertex on the circle, and it intercepts half of the circle so that A = 180/ 2 = 90.

Example 1

What arc length is intercepted by an inscribed angle of 42 on a circle with r = 12 (where = 3.14 = 22/7)?

Solution
The 42 inscribed angle intercepts 1/2(arc) or arc = 84; that is, 84/360 of the circle is intercepted by the angle. The circumference is 2r = 24 so that the arc length is, using = 22/7,

arc length = 84/360 x 24 =
factor out the 12s in 84 and 360, factor 24 into 6 and 4, and convert into 22/7.
(7 x 12)/(30 x 12) x (6 x 4) 22/7 = 88/5 = 17.6

Example 2

A triangle is inscribed in a circle with shorter sides 6 and 8 units long. If the longer side is a diameter, find the length of the diameter.

Solution
A triangle so inscribed (with one side a diameter) is a right triangle. Consequently,

d = 6 + 8 = 36 + 64 = 100; therefore d = 10.

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