Unit 3C: Proportions and similar triangles

Ratio

A comparison of two quantities, written "a to b" or "a:b" or "a/b"

Extended Ratio

used to compare three or more quantities, written "a:b:c"

Proportion

An equation stating that two ratios are equal, the product of the means equals the product of the extremes

Similar Polygons

Polygons that have the same shape, but not necessarily the same size. Corresponding sides are proportional.

scale factor

the ratio of the lengths of two corresponding sides of two similar 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.

Dilation

A transformation that changes the size of an object, but not the shape.

enlargement

a dilation with a scale factor greater than 1 (makes the image larger than the pre-image)

reduction

a dilation with a scale factor between 0 and 1 (makes the image smaller than the pre-image)

Angle-Angle Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Side-Angle-Side 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.

Side-Side-Side Similarity Theorem

If the corresponding sides of two triangles are proportional, then the triangles are similar.

Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

Converse of the Triangle Proportionality Theorem

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

midsegment of a triangle

segment that connects the midpoints of two sides of a triangle

Triangle Midsegment Theorem

A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side

Triangle Proportionality Theorem Corollary

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

Triangle Proportionality Theorem Corollary

If three parallel lines intersect two or more transversals and one of the transversals is cut into congruent parts, then so is every other transversal.

Altitudes of Similar Triangles

The altitudes of similar triangles have the same ratio as the scale factor.

Angle Bisectors of Similar Triangles

The angle bisectors of similar triangles have the same ratio as the scale factor.

Medians of Similar Triangles

The medians of similar triangles have the same ratio as the scale factor.

Triangle Angle Bisector Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides

Scale Model/Drawing

Used to represent objects too large or too small to be drawn/built at actual size