"One can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection."
When two shapes have the same shape (but possibly different sizes), they are said to be similar. This topic covers how to determine if two shapes are similar, how to find scale factors and ratios of similarity, and how to use similarity to solve problems.
Congruence: The relationship between two figures that have the same shape and size.
Scale factor: The ratio of the corresponding sides of two similar figures.
Proportion: The equality of two ratios, usually represented as a fraction.
Similar figures: Two figures that have the same shape but may have different sizes.
Dilations: A transformation that changes the size of a figure while keeping its shape the same.
Isometric transformations: Transformations that preserve length and angle size, such as translation, rotation, and reflection.
Ratios: The comparison of two quantities using division.
Perimeters: The sum of the lengths of the sides of a polygon.
Areas: The measurement of the space inside a shape.
Volumes: The measurement of the amount of space that a solid object occupies.
Pythagorean theorem: A formula that relates the side lengths of a right triangle.
Trigonometry: The study of the relationships between the sides and angles of triangles.
Circles: A shape with all points equidistant from a center point.
Arcs: A portion of the circumference of a circle.
Sectors: A portion of the area enclosed by a circle and the central angle that created it.
Similarity transformations: Transformations that preserve the shape of a figure, such as translations, rotations, reflections, and dilations.
Equations of similar figures: Equations that relate the corresponding sides or angles of similar figures.
Fundamental concepts of similarity: The basics of similarity and how it is used in geometry.
Congruence: Two figures are congruent if they have the same shape and size.
Similarity: Two figures are similar if they have the same shape but not necessarily the same size.
Proportionality: Two figures are proportional if their corresponding sides are in proportion.
Angle similarity: Two figures are angle similar if their corresponding angles are equal.
Scale factor: The ratio of the lengths of corresponding sides of two similar figures is called their scale factor.
Transitivity: If figure A is similar to figure B, and figure B is similar to figure C, then figure A is similar to figure C.
Reflection: Two figures are reflection similar if they are mirror images of each other.
Translation: Two figures are translation similar if one can be obtained from the other by moving it up, down, left, or right.
Rotation: Two figures are rotation similar if one can be obtained from the other by rotating it about a fixed point.
Dilation: Two figures are dilation similar if one can be obtained from the other by enlarging or reducing it uniformly.
"Yes, if one has the same shape as the mirror image of the other."
"Objects can be rescaled, repositioned, and reflected to coincide precisely."
"Yes, all circles are similar to each other."
"No, ellipses are not all similar to each other because they can have different width to height ratios."
"No, rectangles are not all similar to each other as they can have different length to breadth ratios."
"No, isosceles triangles are not all similar to each other as they can have different base angles."
"If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar."
"Corresponding sides of similar polygons are in proportion."
"Yes, corresponding angles of similar polygons have the same measure."
"Yes, two congruent shapes are similar, with a scale factor of 1."
"Some school textbooks specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar."
"Yes, objects can be uniformly scaled, either enlarged or reduced, to be similar to each other."
"Yes, additional operations such as translation, rotation, and reflection can be used to make two shapes identical."
"No, in addition to having the same shape, objects also need to be rescaled or transformed in a specific way to be considered similar."
"Yes, if two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar."
"Rectangles can be determined to be similar if they have the same length to breadth ratio."
"Yes, all equilateral triangles are similar to each other."
"Yes, reflection is one of the operations that can be used to make two objects similar."
"No, corresponding angles of similar polygons have the same measure."