Topic 1: Ratios and Proportions
Definitions:
A ratio is a comparison of two quantities by division. You can write the ratio of two numbers a and b, where b≠0, in three ways: a/b, a:b, a to b
Extended ratios compare three (or more) numbers: If a:b:c then a:b ; b:c ; a:c
Proportions are equations that set two ratios equal to each other.
Helpful Hints:
Ratios must have the same units and be in simplest form.
Video Examples:
A ratio is a comparison of two quantities by division. You can write the ratio of two numbers a and b, where b≠0, in three ways: a/b, a:b, a to b
Extended ratios compare three (or more) numbers: If a:b:c then a:b ; b:c ; a:c
Proportions are equations that set two ratios equal to each other.
Helpful Hints:
Ratios must have the same units and be in simplest form.
Video Examples:
Topic 2: Similar Polygons
Definitions:
Two polygons are similar polygons if corresponding angles are congruent and correspond sides are proportional.
Similarity Statement: ABCD ~ EFGH. This means that:
<A=<E, <B=<F, <C=<G, <D=<H and
AB/EF = BC/FG = CD/GH = DA/HE
Scale Factor: - If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor.
Helpful Hints:
To determine whether two polygons are similar, they must have congruent angles and proportional sides; so check the ratios of the sides to determine if the sides are all proportional.
Video Examples:
Two polygons are similar polygons if corresponding angles are congruent and correspond sides are proportional.
Similarity Statement: ABCD ~ EFGH. This means that:
<A=<E, <B=<F, <C=<G, <D=<H and
AB/EF = BC/FG = CD/GH = DA/HE
Scale Factor: - If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor.
Helpful Hints:
To determine whether two polygons are similar, they must have congruent angles and proportional sides; so check the ratios of the sides to determine if the sides are all proportional.
Video Examples:
Topic 3: Dilations and Scale Factors
Definitions:
A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.
Dilations can come in the form of enlargement, or reduction.
The scale factor determines the size.
Helpful Hints:
Scale Factor for a dilation is always new/old
A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.
Dilations can come in the form of enlargement, or reduction.
The scale factor determines the size.
Helpful Hints:
Scale Factor for a dilation is always new/old
Video Examples:
Topic 4: AA, SAS, SSS
Definitions:
Angle Angle Similarity (AA~): If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
Side Angle Side Similarity (SAS~): If an angle of one triangle is congruent to the angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar
Side Side Side Similarity (SSS~): If the corresponding side lengths of two triangles are proportional, then the triangles are similar
Helpful Hints:
Don't forget about vertical angles being congruent. Those are often not marked.
Sometimes triangles overlap and share and angle or a side.
Video Examples:
Topic 5: Right Triangle Similarity
Definitions:
The Geometric Mean is a special type of average where we multiply the numbers together and then take a square root (for two numbers), cube root (for three numbers) etc.
Helpful Hints:
Redraw the triangles.
The Geometric Mean is a special type of average where we multiply the numbers together and then take a square root (for two numbers), cube root (for three numbers) etc.
Helpful Hints:
Redraw the triangles.
Video Examples:
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Topic 6: Other Proportionality Theorems
Definitions:
Triangle Proportionality Theorem (Side-Splitter Theorem): If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional lengths.
If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.
Angle Bisector Theorem: An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.
Video Examples:
Triangle Proportionality Theorem (Side-Splitter Theorem): If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional lengths.
If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.
Angle Bisector Theorem: An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.
Video Examples: