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Similarity


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ratio
A comparison of two quantities, also know as fancy word for fraction.
proportion
An equation that states that two ratios are equal.
Extremes
the first and last numbers in a proportion
Means
the middle two numbers in a proportion
extended ratio
compares three or more numbers
Cross Products Property
in a proportion, the product of the extremes equals the product of the means
similar figures
Figures that have the same shape, but not necessarily the same size.
Side-Angle-Side Similarity (SAS∼)
two triangles are said to be similar when two pairs of side lengths are proportional and the included angles are congruent
geometric mean
the square root of the product of the extremes or means
Properties of Proportions
If a ÷ b = c ÷ d, then b ÷ a = d ÷ c;
scale factor
the ratio of corresponding linear measurements of two similar figures
extended proportion
when three or more ratios are equal
Angle-Angle Similarity (AA∼)
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Side-Side-Side Similarity (SSS∼)
If the corresponding sides of two triangles are proportional, then the triangles are similar.
indirect measurement
methods of determining lengths that are difficult to measure
The Altitude of a Right Triangle Theorem
The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.
Corollary 1 to the Altitude of a Right Triangle Theorem
The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.
Side-Splitter Theorem
If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.
Corollary to the Side-Splitter Theorem
If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.
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 two sides of the triangle.
Similar
figures that have the same shape but not necessarily the same size; corresponding angles are congruent, but not necessarily corresponding sides
Congruent
A plane figure with the same size and shape.
Dilation
a transformation that changes the size of an object, but not the shape
Pre-image
the original figure BEFORE a transformation is performed on it
image
the resulting figure AFTER a transformation is performed on it
enlargement
dilation whose image is larger than its preimage; |r| > 1
reduction
dilation whose image is smaller than its preimage;