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Geometry Chapter 7
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Gravity
Terms in this set (44)
ratio
a method of comparing two numbers by division, a fraction, must be in simplest form, same unit of measurement
ratio
can't have zero as a denominator, numerator and denominator must be whole numbers
extended ratio
comparing more than two numbers, a:b:c:d
ratio can be...
less than, greater than, or equal to 1 (quantities compared are equal)
common factor is there
ratio must be reduced when....
...
proportion and ratio go hand in hand
proportion
a proportion is an equation setting two ratios equal
means
2nd and 3rd terms of a proportion
extremes
1st and 4th terms of a proportion
extended proportion
sets more than two ratios equal, a/b:c/d:e/f:g/h
proportion
equation=
equivalent equations
equations with the same solution, are formed by adding, subtracting, multiplying, or dividing both sides of an equation by the same number
first property of proportions
a/b=c/d to b/a=d/c
second property of proportions
a/b=c/d is equivalent to a/c=b/d
third property of proportions
a/b=c/d is equivalent to a+b/b=c+d/d
fourth property of proportions
a/b=c/d is equivalent to d/b=c/a
cross products property
in any proportion, the product of the means is equal to the products of the extremes (cross multiplication)
cross multiplication
a useful way to solve for the missing term in a proportion
...
proportions with the same cross products are equivalent
similar polygons
polygons with the same number of sides, same shape but a different size
corresponding angles are congruent, corresponding sides are proportional
...
similar polygons are named according to corresponding vertices
...
number of sides of a polygon determines shape, the length of sides determines size
scale factor
ratio of the lengths of the corresponding sides of similar polygon, ratio of corresponding linear measurements
thrm
if the polygons are similar, then the ratio of the perimeters is equal to the scale factor
Problem solving strategy
1. state the similarity
2. write an extended proportion based off of the statement
3. substitute into the extended proportion
scale drawing
all lengths are proportional to the corresponding actual lengths
easiest way to check if proportion is equivalent
cross products are equal
AA similarity postulate
if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar, remember: if two angles are congruent then the third is as well
SAS similarity theorem
if an angle of one triangle is congruent to an angle of another triangle, and if the sides including those triangles are in proportion, then the triangles are similar
SSS similarity theorem
if the sides of two triangles are in proportion then the two triangles are similar
...
pair big sides with big sides and small sides with small sides
ways to prove 2 triangles are similar
AA, SAS, SSS
how to solve a quadratic equation with no linear term
1. set side equal to zero
2. get the two binomials and factor
3. set each factor equal to zero
OR
1. isolate the x term
2. find the square root of both
Geometric Mean
if x is a positive number such that a/x=x/b, then x is called this
3 Altitudes in a right triangle
legs are 2 of the altitudes and one altitude is from the vertex perpendicular to the hypotenuse
Right Triangle Similar Thrm
if the altitude is drawn to the hypotenuse in a right triangle, then the two triangles formed are similar to the given triangle and eachother
Altitude Corollary
the altitude is a geometric mean between the segments of the hypotenuse
leg corollary
the leg is a geometric mean between the hypotenuse and the segment of the hypotenuse next to the leg
defn
two segments are divided proportionally if the 3 proportions can be written
using properties of proportions
if one proportion is true, then they are all true
Side Splitter Theorem
if a line is parallel to one side of a triangle and intersects the other 2 sides, then it divides those two sides proportionally, says that ratios on one side of a triangle matches ratios on the other side, mid segment does not apply, ratio of the parallel segments is the same as the scale factor of the similar triangles,
method of side splitter theorem
1. write a ratio with x in it
2. put x in a ratio with numbers on the same side
3. match parts on other side
converse side splitter theorem
if two sides of a triangle are divided proportionally, then there is a line that intersects two sides that is parallel to one side of a triangle
triangle-angle bisector theorem
if a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other 2 sides of the triangle
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