Download definitions and theorems 5 - The Bronx High School of Science

Bronx High School of Science
Ms. Abbott
Mathematics Department
M$4
Unit 5: Ratio and Proportion
DEFINITIONS
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a
. (Also written 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 proportion (1st, 2nd, 3rd, and 4th terms, respectively)
The extremes are the first and fourth terms of a proportion.
The means are the second and third terms of a proportion.
If the means of a proportion are equal, it is called the mean proportional.
Two polygons are similar if there is a one-to-one correspondence between the
vertices such that: 1) the measures of all the angles are congruent; 2) the ratio of
the lengths of corresponding sides are equal. (The 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 one 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.