Difference between revisions of "Proportion"
(→Introductory) |
(→Introductory) |
||
| Line 32: | Line 32: | ||
==Problems== | ==Problems== | ||
===Introductory=== | ===Introductory=== | ||
| − | [[Circle]]s with [[center]]s <math>A</math> and <math>B</math> have [[radius |radii]] 3 and 8, respectively. A common internal [[tangent]] [[intersect]]s the circles at <math>C</math> and <math>D</math>, respectively. [[Line]]s <math>AB</math> and <math>CD</math> intersect at <math>E</math>, and <math>AE=5</math>. What is <math>CD</math>? ([[2006 AMC 12A Problems/Problem 16]]) | + | [[Circle]]s with [[center]]s <math>A</math> and <math>B</math> have [[radius |radii]] 3 and 8, respectively. A common internal [[Tangent line|tangent]] [[intersect]]s the circles at <math>C</math> and <math>D</math>, respectively. [[Line]]s <math>AB</math> and <math>CD</math> intersect at <math>E</math>, and <math>AE=5</math>. What is <math>CD</math>? ([[2006 AMC 12A Problems/Problem 16]]) |
===Intermediate=== | ===Intermediate=== | ||
===Pre-Olympiad=== | ===Pre-Olympiad=== | ||
===Olympiad==== | ===Olympiad==== | ||
Revision as of 21:01, 14 September 2007
| This is an AoPSWiki Word of the Week for Sep 13-19 |
Two numbers are said to be in proportion to each other if some numeric relationship exists between them. There are several types of proportions, each defined by a separate class of function.
Contents
Direct proportion
Direct proportion is a proportion in which one number is a multiple of the other. Direct proportion between two numbers x and y can be expressed as:
where k is some real number.
The graph of a direct proportion is always linear.
Often, this will be written as 
.
Inverse proportion
Inverse proportion is a proportion in which as one number's absolute value increases, the other's decreases in a directly proportional amount. It can be expressed as:
where k is some real number that does not equal zero.
The graph of an inverse proportion is always a hyperbola, with asymptotes at the x and y axes.
Exponential proportion
A proportion in which one number is equal to a constant raised to the power of the other, or the logarithm of the other, is called an exponential proportion. It can be expressed as either:
or
or
for some real number k, where k is not zero or one.
Problems
Introductory
Circles with centers 
and 
have radii 3 and 8, respectively. A common internal tangent intersects the circles at 
and 
, respectively. Lines 
and 
intersect at 
, and 
. What is ? (2006 AMC 12A Problems/Problem 16)
