知识点 / GCSE / Mathematics / Geometry and measures Congruence and similarity 边玩边学
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课程笔记 Congruent shapes Shapes are congruent if they are exactly the same shape and size (one can be rotated, reflected or translated onto the other). Triangles are congruent if one of these holds: SSS (three sides equal), SAS (two sides and the included angle), ASA or AAS (two angles and a side), RHS (right angle, hypotenuse and a side). AAA is not enough : it gives similar triangles, not congruent ones.Congruent triangles can be turned Congruent triangles can be turned 4 cm 3 cm 5 cm 4 cm 3 cm 5 cm Same corresponding lengths; orientation may change. Similar shapes Shapes are similar if corresponding angles are equal and corresponding sides are in the same ratio — the scale factor . Scale factor = new length ÷ original length. A rectangle 3 × 5 enlarged to 6 × 10 has scale factor 2. Similar triangles: length factor 2 Similar triangles: length factor 2 4 cm 3 cm 5 cm 8 cm 6 cm 10 cm Corresponding lengths are multiplied by 2. Areas and volumes of similar shapes If the length scale factor is k, then area scales by k2 and volume scales by k3 . Scale factor 3: lengths × 3, areas × 9, volumes × 27. Ratio of areas = (ratio of lengths)²; ratio of volumes = (ratio of lengths)³. Lengths, areas and volumes Lengths, areas and volumes Length factor Area factor Volume factor 2 4 8 3 9 27
Using similar triangles Set up equal ratios of corresponding sides: if triangle ABC ~ DEF with AB = 4, DE = 6, BC = 5, then EF = 5 \times \frac{6}{4} = 7.5. Parallel lines cut by transversals create similar triangles, with equal corresponding angles. 幻灯片 Sign up free to view the lesson slides
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练习题 免费预览——31 题中的 8 题。注册即可查看全部。 1. What does 'congruent' mean for two shapes?
Easy A same shape and sizeB same shape, different sizeC same area onlyD same perimeter only2. Which pair of shapes is always similar?
Easy A any two rectanglesB any two circlesC any two trianglesD any two rhombuses3. What is the scale factor of an enlargement from a 4 cm side to a 12 cm side?
Easy 4. Triangle ABC has sides 3, 4, 5 and DEF has sides 6, 8, 10. Are they similar?
Easy A yes, scale factor 2B noC yes, scale factor 3D congruent5. Which condition proves two triangles congruent: two sides and the angle between them?
Easy 6. Which condition proves two right-angled triangles congruent using the hypotenuse and one side?
Easy 7. A rectangle 3 cm by 5 cm is enlarged by scale factor 2. What are its new dimensions?
Easy A 5 by 7B 6 by 10C 9 by 15D 6 by 58. The scale factor is 3. By what factor is the area multiplied?
Easy 更多题目 9. The scale factor is 2. By what factor is the volume multiplied?
Easy 10. Two similar triangles have corresponding sides 4 cm and 6 cm. The smaller has a side of 5 cm. What is the corresponding side in the larger?
Medium 11. Two similar shapes have areas 20 cm² and 45 cm². Find the length scale factor from the smaller to the larger.
Medium A \frac{3}{2}B \frac{9}{4}C \frac{4}{9}D 2512. Corresponding lengths of two similar shapes have ratio 2:5 in the stated order. What is their area ratio in that same order?
Medium A 4 : 25B 2 : 5C 8 : 125D 4 : 1013. Two similar cylinders have heights 4 cm and 6 cm. The smaller holds 80 ml. What does the larger hold?
Medium A 270 mlB 120 mlC 180 mlD 360 ml14. A model car is built at scale 1 : 20. The real car is 4 m long. How long is the model?
Medium 15. Triangles ABC and DEF are similar with AB = 4, DE = 6 and BC = 5. What is EF?
Medium 16. Two similar prisms have volumes 64 cm³ and 125 cm³. What is the ratio of their surface areas?
Medium A 16 : 25B 4 : 5C 64 : 125D 8 : 1517. A shape has area 30 cm². It is enlarged with scale factor 3. What is its new area?
Medium A 90 cm²B 180 cm²C 270 cm²D 900 cm²18. The volume ratio of two similar solids is 8 : 27. What is the ratio of their heights?
Medium A 2 : 3B 4 : 9C 8 : 27D 64 : 72919. Why are two triangles with all angles equal not necessarily congruent?
Medium A they could be different sizesB angles do not matterC they must be congruentD angles must differ20. Which of these is NOT a transformation that preserves congruence?
Easy A rotationB reflectionC translationD enlargement by 221. In similar triangles, which sides correspond?
Medium A opposite equal anglesB the longest to the shortestC any two sidesD the base only22. Two similar rectangles have widths 3 cm and 9 cm. The smaller has length 5 cm. What is the larger's length?
Medium 23. If lengths are doubled, the area is multiplied by:
Easy 24. Congruent triangles are the same shape and size.
Easy True or false?
25. AAA is a condition for congruent triangles.
Easy True or false?
26. If the length scale factor is 2, the area scale factor is 4.
Easy True or false?
27. Similar shapes always have equal side lengths.
Easy True or false?
28. If the length scale factor is 3, the volume scale factor is 9.
Medium True or false?
29. Which of these prove two triangles congruent?
Medium A SSSB SASC AAAD ASAE SSA in general30. Which are true for similar shapes?
Medium A corresponding angles are equalB sides are in the same ratioC they must be the same sizeD areas are in the ratio k2 E volumes are in the ratio k2 31. Two similar solids have heights 5 cm and 10 cm. The smaller has volume 40 cm³. Find the volume of the larger.
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