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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 ,
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:
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ACB = 1/2 arcAB. |
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A line that just touches a circle is called a tangent. It is perpendicular to the radius drawn to the point of touching. |
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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°. |
What arc length is intercepted by an inscribed angle of 42° on a circle with r = 12 (where = 3.14 = 22/7)?
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.
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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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A certain clock has a minute hand that is
exactly 3 times as long as it's hour hand. Point C is at the tip of
the minute hand, and point D is at the tip of the hour hand. What
is the ratio of the distance that point C travels to the distance
that point D travels in 6 hours?
A. 3:1
B. 6:1
C. 12:1
D. 18:1
E. 36:1
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Solution
In 6 hours, the point C on the minute
hand travels 6 circumferences (where point C to the middle of
the clock is the radius of it's circle). The point D on the
hour hand only travels half way round the clock, half a circumference
(where point D to the middle of the clock is the radius of it's
circle).
Since the minute hand is 3 times as long
as the hour hand, let the distance between point C and the center
of the clock be 3r and the distance from point D to the center
be r.
Point C travels 6 * 2 ¹ (3r) = 36 ¹ r
Point D travels 0.5 * 2 ¹ r = ¹ r
Thus, the ratio of the distance that
point C travels to the distance that point D travels in 6 hours
is 36:1. The correct answer is E.
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