Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Dessin d'enfant wikipedia , lookup
Penrose tiling wikipedia , lookup
Multilateration wikipedia , lookup
Technical drawing wikipedia , lookup
Tessellation wikipedia , lookup
Golden ratio wikipedia , lookup
Perceived visual angle wikipedia , lookup
Rational trigonometry wikipedia , lookup
Euler angles wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Similarity: Key Terms Term Transformation Definition A change to a geometric shape using various mathematical criteria. Examples include: Turning (formal math word: Rotation) Flipping (formal math word: Reflection) Sliding (formal math word: Translation) Resizing (formal math word: Dilation) Dilation A dilation is a change to a geometric shape, based on multiplying the side lengths by a defined scale factor. Dilations preserve the angles in a shape, but change the side lengths, so a similar figure is created. Center of dilation The fixed point around which all the points in the figures are expanded / contracted. The center of dilation is NOT necessarily the center of the shape! (If you’re an art person, you can think of the center of dilation as the “vanishing point” of the figures.) Example Scale factor Ratio Proportion The scale factor is the constant number which multiplies the side lengths of the original shape to create the dilated shape. A ratio is a statement of how two numbers compare to each other. There are several ways to write ratios. 2 As a fraction 3 With a colon 2:3 In words “2 to 3” We can also write EQUIVALENT ratios (meaning, the relationship between the numbers is the same). For 2 4 10 200 instance, some equivalent ratios to 3 could be 6 , 15 , 300, etc. A proportion is a statement about the equality of two or more ratios. Triangles ABC and WTF are similar. You can see that triangle WTF is ½ the size of triangle ABC, so the scale factor for the dilation of triangle ABC to get triangle WTF is ½. (Note: if we wanted to dilate triangle WTF to get triangle ABC, we would have a scale factor of 2 because triangle ABC is twice the size of triangle WTF.) Let’s say there are 10 girls and 12 boys in a typical 4th grade class. We could say the ratio of girls to boys is: “10 to 12” 10:12 10 Or 12 We could also write equivalent ratios: “5 to 6” 5:6 5 Or 6 Let’s use the example in the definition above. We can write the following proportion: 2 4 10 200 = = = 3 6 15 300 Congruent Two shapes that are congruent are the same shape AND size. This means that the angles AND side lengths of the shape are exactly the same. The following pentagons are congruent (same shape, same size). We can also refer to line segments, angles, etc. as congruent. Congruent line segments are the same length, and congruent angles are the same measurement (same “width”). Congruent shapes do NOT have to be oriented in the same direction – they can be flipped or rotated. Similar Two shapes that are similar are the same shape, but may NOT be the same size. (Note that congruent shapes can also be defined as similar). The following quadrilaterals are similar (same shape, different size – note that they are NOT oriented the same). Similar shapes have congruent corresponding ANGLES, and proportional corresponding SIDES. Similar shapes do NOT have to be oriented in the same direction – they can be flipped or rotated. Polygon A polygon is a straight-sided shape with at least 3 sides. Polygons can be regular (all sides the same length) or irregular (different sides of different lengths). All of the straight-sided shapes in previous examples are polygons. Corresponding Corresponding parts of a shape are the parts that “match” based on their position relative to the entire shape. Angle-Angle postulate for triangle similarity The Angle-Angle similarity postulate says: IF two triangles have two pairs of corresponding angles which are congruent, THEN the triangles are similar. (Note: this only works for triangles!) Side-Angle-Side postulate for triangle similarity The Side-Angle-Side similarity postulate says: IF two triangles have 2 corresponding sides proportional AND the angles between them are congruent, THEN the triangles are similar. (Note: this only works for triangles!) Angle A and Angle Z are corresponding angles in these similar figures. Line segments AT and ZL are corresponding sides in these similar figures. Triangles FUN and SUX are similar by Angle-Angle. Angle NFU and Angle XSU are labeled as congruent; also, Angle SUX has to be congruent with Angle FUN because they are the same angle. Triangles ABC and FGH are similar by Side-AngleSide. The proportion of the corresponding side 16 40 lengths: 12 = 30 is true, and the angles included between them are congruent. Side-Side-Side postulate The Side-Side-Side similarity postulate says: IF two triangles have 3 proportional pairs of corresponding sides, THEN the triangles are similar. Triangles CAT and DOG are similar by Side-Side-Side. All 3 pairs of corresponding sides are proportional. 3.5 3.5 5 = = 7 7 10