Geometry Chapter 7 Flashcards

Ratios and Proportions

• A ratio compares two quantities of the same unit, expressed as a/b or a : b, where b ≠ 0.
• Equivalent ratios maintain the same simplified form.
• A proportion consists of two equal ratios.
• In a proportion, the extremes are the first and last terms (a and d), while the means are the middle terms (b and c).
• The cross product property states that in a proportion, the product of the extremes equals the product of the means: ad = bc.

Geometric Mean

• The geometric mean between two positive numbers a and b is a positive number x such that a/x = x/b.
• Algebraically, x^2 equals the product of the two numbers: x^2 = ab, and x represents the square root of ab.

Triangle Similarity

• Two polygons are similar if their corresponding angles are congruent and the lengths of corresponding sides are proportional.
• The symbol ~ denotes similarity.
• The scale factor is the ratio of the lengths of two corresponding sides in similar polygons.
• Ratios of perimeters of similar polygons are equal to the ratios of their corresponding sides.

Similarity Postulates and Theorems

• AA similarity postulate: Two triangles are similar if two angles of one triangle are congruent to two angles of another triangle.
• SSS similarity theorem: If corresponding sides of two triangles are proportional, they are similar.
• SAS similarity theorem: If an angle of one triangle is congruent to an angle of another, and the lengths of included sides are proportional, the triangles are similar.

Properties of Similarity

• Reflexive property: Triangle ABC is similar to itself (ABC ~ ABC).
• Symmetric property: If triangle ABC is similar to triangle DEF, then DEF is similar to ABC (ABC ~ DEF implies DEF ~ ABC).
• Transitive property: If triangle ABC ~ triangle DEF and triangle DEF ~ triangle XYZ, then ABC ~ XYZ.

Proportionality in Triangles

• Triangle proportionality theorem: A line parallel to one side of a triangle intersects the other sides, dividing them proportionally.
• Triangle proportionality converse: If a line divides two sides proportionally, it is parallel to the third side.
• The midsegment connects midpoints of two sides and is parallel to the third side while being half its length.

Proportional Segments

• If three parallel lines intersect two transversals, they divide the transversals proportionally.
• If three or more parallel lines cut off congruent segments on one transversal, they create congruent segments on all transversals.
• The altitude of a triangle creates perpendicular lines from a vertex to the opposite side.

Corresponding Lengths in Similar Triangles

• The lengths of corresponding altitudes in similar triangles are proportional to the lengths of corresponding sides.
• The lengths of angle bisectors in similar triangles correspond proportionally to side lengths.
• The lengths of medians in similar triangles follow the same proportionality as the sides.

Angle Bisectors and Segments

• An angle bisector in a triangle divides the opposite side into segments that are proportional to the lengths of the other two sides.