similarity Flashcards
(20 cards)
Q: Is this correct?
“If a/b = c/d, then ad = bc.”
A: Yes, this is correct.
Explanation: This is the fundamental property of proportions. If two ratios are equal (a/b = c/d), then the cross products are equal, meaning ad = bc.
Name: Reflexive Property of Similarity
Definition: A figure is similar to itself.
Application: This property is used to show that any figure is similar to itself in geometry, especially when proving similarity relations between multiple figures.
Q: Is this correct?
“If a/b = c/d, then b/a = d/c.”
A: Yes, this is correct.
Explanation: This is the inverse property of proportions. If a/b = c/d, then swapping the numerators and denominators gives b/a = d/c.
Name: Transitive Property of Similarity
Definition: If figure A is similar to figure B and figure B is similar to figure C, then figure A is similar to figure C.
Application: This property helps in proving that if multiple figures are similar to each other, then they are all similar.
Q: Is this correct?
“If a/b = c/d, then a/(a+b) = c/(c+d).”
A: No, this is not correct.
Explanation: This is not a standard property of proportions. Proportions of the form a/b = c/d do not necessarily lead to this kind of relationship involving addition of the terms.
Name: Parallel Line → Similar Triangles
Definition: If a line is parallel to a side of a triangle and intersects the other two sides, then the smaller triangle formed is similar to the larger triangle.
Application: This property is used in triangle similarity proofs and geometric problems where parallel lines divide triangles proportionally.
Q: Is this correct?
“If two triangles have three pairs of sides in the same ratio, then the triangles are similar.”
A: Yes, this is correct.
Explanation: This is the SSS (Side-Side-Side) Similarity Theorem. If the sides of two triangles are in proportion, the triangles are similar.
Name: SAS Similarity Theorem
Definition: If two triangles have two pairs of sides in the same ratio and the included angle of those sides is also congruent, then the triangles are similar.
Application: This theorem is useful for proving triangle similarity when two sides and the included angle are given.
Q: Is this correct?
“If two triangles have two pairs of sides in the same ratio and the included angle is different, then the triangles are similar.”
A: No, this is not correct.
Explanation: The SAS Similarity Theorem requires that the included angle between the two sides be congruent. If the angles are not congruent, the triangles may not be similar.
Name: Perimeters of Similar Polygons
Definition: If two polygons are similar, then the ratio of their perimeters is the same as the ratio of their corresponding sides.
Application: This property helps solve problems involving the perimeters of similar polygons, where the ratio of the perimeters is proportional to the ratio of the sides.
Name: Altitude to Hypotenuse Theorem
Definition: If a segment is the altitude to the hypotenuse of a right triangle, then the resulting two triangles are similar to each other and to the original right triangle.
Application: This theorem is used when analyzing right triangles and their properties related to altitudes and similarity.
Q: Is this correct?
“If a segment is the altitude to the hypotenuse of a right triangle, then its length is the arithmetic mean of the lengths of the two segments of the hypotenuse it creates.”
A: No, this is not correct.
Explanation: The length of the altitude is the geometric mean, not the arithmetic mean, of the two segments of the hypotenuse.
Name: Geometric Mean Theorem
Definition: If a segment is the altitude to the hypotenuse of a right triangle, then its length is the geometric mean of the lengths of the two segments of the hypotenuse it creates.
Application: This theorem is used to find the length of the altitude in a right triangle when dealing with the hypotenuse.
Q: Is this correct?
“If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally.”
A: Yes, this is correct.
Explanation: This is the Side Splitter Theorem, which shows that a line parallel to one side of a triangle divides the other two sides proportionally.
Name: Converse of Side Splitter Theorem
Definition: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
Application: This converse is used to prove that a line dividing the sides of a triangle proportionally is parallel to the third side.
Q: Is this correct?
“If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.”
A: Yes, this is correct.
Explanation: This is the Two Transversal Proportionality Theorem, which states that parallel lines dividing two transversals create proportional segments.
Name: Angle Bisector Proportionality Theorem
Definition: If a segment is an angle bisector of an angle in a triangle, then it divides the opposite sides into two segments that are proportional to the lengths of the other two sides.
Application: This theorem is used when dealing with angle bisectors in triangles to determine proportionality between sides.
Q: Is this correct?
“Two polygons are similar if their corresponding angles are congruent AND their corresponding side lengths are proportional.”
A: Yes, this is correct.
Explanation: This is the definition of similarity for polygons, where corresponding angles must be congruent, and corresponding sides must be in proportion.
Name: Geometric Mean
Definition: A number is the geometric mean of two numbers if it is the square root of the product of the two numbers.
Application: The geometric mean is often used in problems involving proportionality, altitudes in right triangles, or other geometric contexts.
Name: AA ~ Postulate
Definition: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Application: This postulate is a common way to prove triangle similarity by showing that two corresponding angles are congruent.
