Congruence And Similarity

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Congruence

Two shapes are congruent if they are identical in shape and size.
One shape may be a reflection, rotation, or translation of the other.
If one shape is an enlargement of the other, they are not congruent.
To prove congruence, show that corresponding sides are equal in length and corresponding angles are equal in size.
Tracing paper can help check congruence if shapes are drawn to scale.

Congruent Shapes

Similarity

Two shapes are similar if they have the same shape and their corresponding sides are in proportion.
One shape is an enlargement of the other; similarity does not imply congruence.
For triangles, prove similarity by showing corresponding angles are equal (e.g., using vertically opposite angles, alternate angles on parallel lines).
For non-triangular shapes, show that all corresponding sides are in the same ratio (scale factor).
If all angles are equal, the shapes are similar (but not necessarily congruent).

Similar Triangles in an Hourglass

Similar Lengths

Equivalent lengths on similar shapes are linked by a scale factor (k).
If the second shape is larger, k > 1; if smaller, 0 < k < 1.
To find k: divide a length on the second shape by the corresponding length on the first shape.
To find a missing length: multiply the corresponding length by k (if going to larger) or divide by k (if going to smaller).
Redraw overlapping similar shapes separately to avoid confusion.

Similar Rectangles

Similar Areas & Volumes

If length scale factor = k, then area scale factor = k² and volume scale factor = k³.
Given area scale factor, length scale factor = √(area SF); volume scale factor = (√(area SF))³.
Given volume scale factor, length scale factor = ∛(volume SF); area scale factor = (∛(volume SF))².
To find missing area or volume: identify known quantities, find the relevant scale factor, then multiply or divide accordingly.
Always check whether the result should be larger or smaller than the given quantity.

Similar Solids: Length, Area and Volume Scale Factors

Worked Examples (Lengths)

Example: Two similar rectangles have sides 6 cm and 3 cm. Scale factor = 6/3 = 2 (larger to smaller) or 3/6 = 0.5 (smaller to larger).
To find missing side: if AD = 15 cm on larger, corresponding PS on smaller = 15 × 0.5 = 7.5 cm.
Always identify corresponding sides correctly before applying scale factor.

Finding a Missing Length on a Similar Shape

Worked Examples (Areas & Volumes)

Example: Solid A volume 32 cm³, solid B volume 108 cm³. k³ = 108/32 = 27/8, so k = ∛(27/8) = 3/2.
If height of A = 10 cm, height of B = 10 × (3/2) = 15 cm.
For areas: if area scale factor = 4, then length scale factor = √4 = 2.

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Câu hỏi luyện tập

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1.Two shapes are congruent if they are identical in shape and size. Which of the following transformations does NOT change congruence?
Easy
- AEnlargement
- BReflection
- CRotation
- DTranslation
2.Triangle ABC is similar to triangle PQR. AB = 6 cm, PQ = 9 cm. What is the length scale factor from triangle ABC to triangle PQR?
Easy
- A1.5
- B0.666...
- C2
- D3
3.Two rectangles are similar. The smaller rectangle has width 4 cm and length 6 cm. The larger rectangle has width 10 cm. What is the length of the larger rectangle?
Easy
- A15 cm
- B12 cm
- C20 cm
- D8 cm
4.Solid A and solid B are mathematically similar. The volume of solid A is 32 cm³ and the volume of solid B is 108 cm³. The height of solid A is 10 cm. Find the height of solid B.
Medium
- A15 cm
- B20 cm
- C12 cm
- D18 cm
5.Two cones are mathematically similar. The total surface area of the smaller cone is 80 cm² and of the larger cone is 180 cm². The volume of the smaller cone is 168 cm³. Calculate the volume of the larger cone.
Hard
- A567 cm³
- B378 cm³
- C756 cm³
- D283.5 cm³
6.A model of a car has a scale 1:20. The volume of the actual car is 12 m³. Find the volume of the model in cubic centimetres.
Medium
- A1500 cm³
- B1500000 cm³
- C3000 cm³
- D600 cm³
7.Two mathematically similar containers have heights of 30 cm and 75 cm. The larger container has a capacity of 5.5 litres. Calculate the capacity of the smaller container in millilitres.
Medium
- A352 ml
- B880 ml
- C220 ml
- D550 ml
8.Two containers are mathematically similar. The surface area of the larger container is 226 cm² and the surface area of the smaller container is 94 cm². The volume of the larger container is 680 cm³. Find the volume of the smaller container.
Hard
- A183 cm³
- B150 cm³
- C200 cm³
- D250 cm³

Thêm câu hỏi

9.A paperweight has height 4 cm and volume 38.4 cm³. A mathematically similar paperweight has height 7 cm. Calculate the volume of this paperweight.
Medium
- A205.8 cm³
- B117.6 cm³
- C67.2 cm³
- D120 cm³
10.Which of the following statements is true about similar shapes?
Easy
- AThey have the same shape and size.
- BThey have the same shape but not necessarily the same size.
- CThey have the same size but not necessarily the same shape.
- DThey are always congruent.
11.In the diagram, AB and CD are parallel lines. Which pair of triangles are similar?
Easy
- ATriangle ABX and triangle CDX
- BTriangle ABC and triangle ACD
- CTriangle AXD and triangle BXC
- DTriangle ABD and triangle CBD
12.Marianne sells two sizes of photo. These photos are mathematically similar rectangles. The smaller photo has length 15 cm and width 12 cm. The larger photo has area 352.8 cm². Calculate the length of the larger photo.
Hard
- A21 cm
- B18 cm
- C24 cm
- D16.8 cm
13.Two cups A and B are made from a thin plastic and shaped as cones. Cup A has a radius of 3 cm and a height of 12 cm. Cup B is geometrically similar to cup A. The capacity of cup B is 252π cm³ more than cup A. Find the radius of cup B.
Challenge
- A6 cm
- B4.5 cm
- C7.5 cm
- D5 cm
14.The area of pentagon A is 73.5 cm². The area of pentagon B is 6 cm². Find the ratio perimeter of pentagon A : perimeter of pentagon B in its simplest form.
Medium
- A7:2
- B3.5:1
- C49:4
- D7:4
15.Two triangles are similar. Which of the following statements must be true?
Easy
- ATheir corresponding sides are equal in length.
- BTheir corresponding angles are equal.
- CTheir areas are equal.
- DThey are congruent.
16.Two triangles are similar. The sides of the first triangle are 3 cm, 4 cm, and 5 cm. The shortest side of the second triangle is 9 cm. What is the length of the longest side of the second triangle?
Easy
- A12 cm
- B15 cm
- C27 cm
- D20 cm
17.Two similar cylinders have heights 10 cm and 25 cm. The smaller cylinder has a volume of 200 cm³. What is the volume of the larger cylinder?
Medium
- A500 cm³
- B3125 cm³
- C1250 cm³
- D800 cm³
18.A model of a building is made with a scale factor of 1:50. The model has a height of 1.2 m. What is the height of the real building?
Easy
- A60 m
- B6 m
- C50 m
- D120 m
19.Two similar prisms have volumes 648 cm³ and 2187 cm³. The cross-sectional area of the smaller prism is 36 cm². What is the cross-sectional area of the larger prism?
Medium
- A81 cm²
- B108 cm²
- C54 cm²
- D72 cm²
20.Two rectangles are similar. The smaller rectangle has sides 4 cm and 6 cm. The larger rectangle has area 150 cm². What is the length scale factor?
Medium
- A2.5
- B1.5
- C3
- D2
21.Two similar triangles have areas 16 cm² and 25 cm². The base of the smaller triangle is 8 cm. What is the base of the larger triangle?
Medium
- A10 cm
- B12.5 cm
- C9 cm
- D11 cm
22.Two similar shapes have areas 295 cm² and 159.5 cm². The width of the larger shape is 17 cm. What is the width of the smaller shape?
Medium
- A12.5 cm
- B13.5 cm
- C11.5 cm
- D14.5 cm
23.Two similar cylinders: cylinder A has height 1.6 m and radius 0.56 m. Cylinder B has height 0.6 m. What is the radius of cylinder B?
Medium
- A0.21 m
- B0.35 m
- C0.14 m
- D0.28 m
24.A model of a statue has height 4 cm and volume 12 cm³. The statue has volume 40500 cm³. What is the height of the statue?
Medium
- A135 cm
- B60 cm
- C90 cm
- D180 cm
25.If two shapes are similar, then they are always congruent.
Easy
True or false?
26.Two cones are mathematically similar. The total surface area of the smaller cone is 80 cm² and of the larger cone is 180 cm². The volume of the smaller cone is 128 cm³. What is the volume of the larger cone?
Medium
- A192 cm³
- B216 cm³
- C270 cm³
- D432 cm³
27.Match each scale factor type to its relationship with the length scale factor k.
Easy
- Length scale factor
- Area scale factor
- Volume scale factor
- k
- k²
- k³
28.Arrange the steps to find the height of a similar solid given the volumes and the height of the smaller solid.
Medium
- Find the volume scale factor by dividing larger volume by smaller volume.
- Take the cube root of the volume scale factor to get the length scale factor.
- Multiply the smaller height by the length scale factor to get the larger height.
29.Which of the following conditions are sufficient to prove two triangles are similar? (Select all that apply)
Medium
- AAll three corresponding angles are equal.
- BTwo sides are in proportion and the included angle is equal.
- CAll three corresponding sides are in proportion.
- DTwo angles are equal and a non-included side is in proportion.
- EThe triangles have the same shape but different size.
30.Two triangles are similar. Which of the following conditions is sufficient to prove similarity?
Medium
- ATwo pairs of sides are in proportion and the included angles are equal
- BTwo pairs of sides are equal
- COne pair of sides is in proportion and one pair of angles is equal
- DAll three sides are equal
31.Which of the following are true about two mathematically similar solids? (Select all that apply)
Medium
- ACorresponding lengths are in the same ratio
- BCorresponding angles are equal
- CThe ratio of their volumes is equal to the ratio of their corresponding lengths
- DThe ratio of their surface areas is equal to the square of the ratio of their corresponding lengths
32.In the diagram, AB is parallel to CD. Which of the following conditions is sufficient to prove that triangles ABX and CDX are similar?
Medium
- AAX = XC and BX = XD
- BAngle AXB = angle CXD and angle ABX = angle CDX
- CAB = CD and AX = XC
- DAngle ABX = angle XCD and AX = XD
33.Complete the sentence: Two shapes are similar if they have the same shape and their corresponding sides are in ___.
Easy
Two shapes are similar if they have the same shape and their corresponding sides are in ____.
34.In the diagram, AB and CD are parallel lines. Which pair of triangles are similar? (Write the names of the two triangles, e.g., ABX and CDX)
Easy
- ATriangle ABX and triangle CDX
- BTriangle ABC and triangle ACD
- CTriangle AXD and triangle BXC
- DTriangle ABD and triangle CBD
35.Two similar cylinders have volumes 64 cm³ and 216 cm³. The surface area of the smaller cylinder is 80 cm². Find the surface area of the larger cylinder.
Medium
36.Two similar rectangles have widths 4 cm and 10 cm. The area of the smaller rectangle is 24 cm². Find the area of the larger rectangle.
Medium
37.Complete the sentence: Two shapes are congruent if they have the same shape and ___.
Easy
Two shapes are congruent if they have the same shape and ____.
38.Triangle ABC is similar to triangle PQR. AB = 6 cm, BC = 8 cm, AC = 10 cm. The longest side of triangle PQR is 15 cm. Find the length of the shortest side of triangle PQR.
Medium
39.Two cones are mathematically similar. The volume of the smaller cone is 64 cm³ and the volume of the larger cone is 216 cm³. The surface area of the smaller cone is 80 cm². Find the surface area of the larger cone.
Hard
40.Complete the sentences about scale factors.
Easy
If the length scale factor between two similar solids is k, then the area scale factor is ____ and the volume scale factor is ____.
41.In the diagram, AB and CD are parallel lines. Triangles ABX and CDX share vertex X. Prove that triangles ABX and CDX are similar.
Hard
42.Two similar cylinders have volumes 64 cm³ and 216 cm³. The surface area of the smaller cylinder is 80 cm². Find the surface area of the larger cylinder.
Medium
- A198 cm²
- B216 cm²
- C162 cm²
- D180 cm²
43.Two similar rectangles have widths 4 cm and 10 cm. The area of the smaller rectangle is 24 cm². Find the area of the larger rectangle.
Medium
- A135 cm²
- B150 cm²
- C165 cm²
- D180 cm²
44.Triangle ABC is similar to triangle PQR. AB = 6 cm, BC = 8 cm, AC = 10 cm. The longest side of triangle PQR is 15 cm. Find the length of the shortest side of triangle PQR.
Medium
- A8 cm
- B10 cm
- C9 cm
- D11 cm

Ghi chú ôn tập

## Congruence & Similarity ### Congruence - **Congruent**: identical in shape and size. One may be a reflection, rotation, or translation of the other. - To prove: show corresponding sides equal and corresponding angles equal. - Not congruent if one is an enlargement. ### Similarity - **Similar**: same shape, corresponding sides in proportion (one is an enlargement of the other). - For triangles: prove all corresponding angles equal → sides automatically in proportion. - For other shapes: show scale factor same for all corresponding sides. ### Similar Lengths - **Length scale factor (k)**: divide a length on second shape by corresponding length on first. - To find missing length: multiply by k if going to larger shape; divide by k if going to smaller. ### Similar Areas & Volumes - If length scale factor = k, then: - Area scale factor = k² - Volume scale factor = k³ - To convert between scale factors: - k = √(area SF) = ∛(volume SF) - area SF = k² = (∛(volume SF))² - volume SF = k³ = (√(area SF))³ **Exam Tips**: - Sketch triangles separately if they overlap. - Use tracing paper to check congruence (if drawn to scale). - Label shapes clearly to avoid mixing up.

Congruent: Two shapes are congruent if they are identical in shape and size; one may be a reflection, rotation, or translation of the other.
Similar: Two shapes are similar if they have the same shape and their corresponding sides are in proportion; one is an enlargement of the other.
Scale factor (length): The ratio of corresponding lengths on two similar shapes; found by dividing a length on the second shape by the corresponding length on the first.
Area scale factor: The square of the length scale factor; used to relate areas of similar shapes.
Volume scale factor: The cube of the length scale factor; used to relate volumes of similar shapes.

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