Download Unit 5: Ratios and Proportions - The Bronx High School of Science

Bronx High School of Science
M$4
Mathematics Department
Ms. Abbott
Unit 5: Ratios and Proportions
DEFINITIONS:
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a
(or a:b)
b
a c
A proportion is an equation that states that two ratios are equal. (  )
b d
a,b,c,and d are terms of the ratio/proportion (1st, 2nd, 3rd, and 4th terms,
respectively)
a and d (1st and 4th terms) are the extremes of the proportion.
b and c (2nd and 3rd terms) are the means of the proportion.
If the means of two proportions are equal, it is called the mean proportional of
the two extremes.
Two polygons are similar if there is a one-to-one correspondence between the
vertices such that: 1) measures of all the angles are equal; 2) the ratios of the
lengths of corresponding sides are equal. (Corresponding sides of similar triangles
are in proportion.)
The ratio of two numbers, a and b, b  0 , is the number
POSTULATES:
THEOREMS and COROLLARIES
 In a proportion, the product of the means is equal to the product of the extremes.
 In a proportion, the means may be exchanged.
 In a proportion, the extremes may be exchanged.
 If two angles of one triangle are congruent to two angles of another triangle, the
triangles are similar.
 If a line is parallel to one side of a triangle and intersects the other two sides, it
cuts off a triangle similar to the original triangle.
 If a line is parallel to one side of a triangle and intersects the other two sides, it
divides those sides proportionally.
 If a line intersects two sides of a triangle and divides those sides proportionally, it
is parallel to the third side.
 The segment joining the midpoints of two sides of a triangle is parallel to the third
side and has a length equal to half the length of the third side.
 In any right triangle, the altitude to the hypotenuse separates the triangle into two
triangles which are similar to each other and to the original triangle.
 Right Triangle Altitude Theorem: If the altitude in any right triangle is drawn to
the hypotenuse: 1) the altitude is the mean proportional between the segments of
the hypotenuse; 2) each leg is the mean proportional between the entire
hypotenuse and the segment of the hypotenuse adjacent to that leg.