Unit 4: Similarity Flashcards
If a and b are two numbers or quantities and b≠0, then the ratio of a to b is…
There are 2 ways!
a/b (as a fraction)
OR
a:b
Two ratios that have the same simplififed form are called…
Equivalent Ratios
Ratios are usually expressed in simplest form.
An equation that states 2 ratios are equal is called a…
Proportion
Cross Products Property is…
…the product of the extremes equals the products of the means:
extreme -> a/b <– mean = mean –>c/d <–extreme
If a/b = c/d where b≠0 and d≠0, then ad=bc:
2/3 = 4/6 –> 3(4) = 2(6) –> 12 = 12
What is the geometric mean?
The geometric mean of two positives a and b is the positive number x that satisfies:
a/x = x/b –> x^2 = ab –> x = √ab
What is Reciprocal Property?
If two ratios are equal, then their reciprocals are also equal.
If you interchange the means of a proportion, then you for another true proportion.
In a proportion, if you add the value of each ratio’s denominator to its numerator, then you form another true proportion.
A Scale Drawing is…
…a drawing that is the same shape as the object it represents. The SCALE is a ratio that describes how the dimensions in a drawing are related to the actual dimensions of the object.
Two polygons are Similar Polygons if…
…corresponding angles are congruent AND corresponding side lengths are proportional.
“△ABC is similar to △XYZ” is written as “△ABC~△XYZ”
Order of the letters MATTER!! (Just like congruent polygons)
Perimeters of Similar Polygons:
If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.
If △KLM~△PQR, then…
KL+LM+KM/PQ+QR+PR –>
KL/PQ = LM/QR = KM/PR
Corresponding Lengths in Similar Polygons:
If two polygons are similar, then the ratio of any two corresponding lengths is equal to the scale factor of the similar polygon.
Angle-Angle (AA) Similarity Postulate:
If two angles of one triangle are congurent to two angles of another triangle, then the two triangles are similar.
Side-Side-Side (SSS) Similarity Theorem:
If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
If AB/RS = BC/ST = CA/ RT, then △ABC~△RST
Side-Angle-Side (SAS) Similarity Theorem:
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.
If <X≅<M, and ZX/PM = XY/MN, then △XYZ~△MNP
Triangle Similarity Theorem:
If two triangles are similar, then pairs of corresponding sides have the same ratio and pairs of corresponding angles are congruent.
Equilateral Triangle Similarity Theorem:
All equilateral triangles are similar to each other.
If we know that two triangles are congruent, we can conclude that any pair of corresponding parts (sides or angles), often abbreviated as CPCTC. With similar triangles we need to make a distinction. We can say that:
Corresponding —— of similar triangles are congruent.
Corresponding —– of similar triangles are proportional.
Corresponding ANGLES of similar triangles are congruent. (CASTC)
Corresponding SIDES of similar triangles are proportional. (CSSTP)
TIp: Remember abbrevations by remembering words!
Triangle Proportionality Theorem:
If a line parallel to one side of a trangle intersects the other two sides, then it divides the two sides proportionally.
Converse of the Tirangle Proportionality Theorem:
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
Three Parallel Lines Theorem:
If three parallel lines intersect at two transversals, then they divide the transversals proportionally.
Triangle Angle Bisector Theorem:
If a ray bisects an angle of a tringle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.
Right Triangle Similarity Theorem:
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
Proportions Involving Geometric Mean:
length of shorter leg of △ABC/length of shorter leg of triangle △DEF= length of longer leg of △ABC/length of longer leg of △DEF
length of hypotenuse of △GHI/length of hypotenuse of △ABC = length of shorter leg of △GHI/length of shorter leg of △ABC
length of hypotenuse of △GHI/length of hypotenuse of △DEF = length of longer leg of △GHI/length of longer leg of △DEF
Geometric Mean Theorem (ALTITUDES):
In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments.
The length of the altitude is the geometric mean of the lengths of teh two segments.
Geometric Mean Theorem (LEGS):
In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two semgents.
The length of each leg of the right triangle is the geoetric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
