Congruence and Similarity Notes

Congruence and Similarity

Congruent Shapes

  • Definition: Congruent shapes are identical in shape and size but may have undergone transformations such as:

    • Reflections

    • Rotations

Similar Shapes

  • Definition: Similar shapes are shapes that have undergone enlargements. This relationship shows how the shapes maintain proportionality while differing in size.

  • Scale Factor: When a shape is enlarged, it is increased by a value called the scale factor. All lengths of the shape are enlarged by the same ratio.

Example of Scale Factor Calculation

  • Rectangles:

    • Smaller rectangle width: 2 cm

    • Larger rectangle width: 4 cm

    • Scale factor calculation:
      extScaleFactor=racextLargerWidthextSmallerWidth=rac4extcm2extcm=2ext{Scale Factor} = rac{ ext{Larger Width}}{ ext{Smaller Width}} = rac{4 ext{ cm}}{2 ext{ cm}} = 2

    • Thus, the scale factor is 2.

Properties of Similar Triangles

  • Two triangles are similar if:

    • Corresponding angles are equal.

    • If two lengths are enlarged by the same scale and the angle between them is equal.

Non-Similar Triangles Example

  • Triangles Comparison:

    • Smaller Triangle Angles: 40°, 61°

    • Missing Angle Calculation:
      180°−40°−61°=79°180° - 40° - 61° = 79°

    • Larger Triangle Angles: 60°, 57°

    • Missing Angle Calculation:
      180°−60°−57°=63°180° - 60° - 57° = 63°

    • Conclusion: Angles are not equal; therefore, the triangles are not similar.

Area of Similar Shapes

  • Area calculations can be derived from the scale factor of lengths.

  • If the scale factor for length is kk, then the scale factor for area is k2k^2.

Example Area Calculation

  • Given:

    • Area of Rectangle A = 8 cm²

    • Linear Scale Factor Calculation:
      extLinearScaleFactor=rac12extcm4extcm=3ext{Linear Scale Factor} = rac{12 ext{ cm}}{4 ext{ cm}} = 3

    • Area Scale Factor:
      extAreaScaleFactor=(3)2=9ext{Area Scale Factor} = (3)^2 = 9

    • Area of Rectangle B:
      extAreaofB=extAreaofAimes9=8imes9=72extcm2ext{Area of B} = ext{Area of A} imes 9 = 8 imes 9 = 72 ext{ cm²}

Volume of Similar Shapes

  • Volume can also be calculated using the scale factor.

  • If the scale factor for length is kk, the scale factor for volume is k3k^3.

Example Volume Calculation

  • For Cylinder A:

    • Volume = 4 cm³, Radius = 1 cm

  • For Cylinder B (Volume = 108 cm³):

    1. Volume Scale Factor:
      extVolumeScaleFactor=rac108extcm34extcm3=27ext{Volume Scale Factor} = rac{108 ext{ cm}^3}{4 ext{ cm}^3} = 27

    2. Linear Scale Factor:
      k=extVolumeScaleFactorrac13=27rac13=3k = ext{Volume Scale Factor}^{ rac{1}{3}} = 27^{ rac{1}{3}} = 3

    3. Calculating Radius of Cylinder B:
      extRadiusofB=extRadiusofAimesk=1extcmimes3=3extcmext{Radius of B} = ext{Radius of A} imes k = 1 ext{ cm} imes 3 = 3 ext{ cm}

Practice Questions

  1. Given that the rectangles are similar, calculate length xx cm.

  2. Given that the triangles are similar, calculate length xx cm.

  3. Given that the rectangles are similar, calculate length xx cm.

  4. Two isosceles triangles: Are they similar?

    • Triangle A: Sides 3.5 cm, 6.1 cm, 6.5 cm. Angle: 66°66°.

    • Triangle B: Sides 10.5 cm, xx cm, 11 cm. Angles 51°51°.

  5. Calculate the area scale factor for the similar rectangles.

  6. Given the surface area of cube A (24 cm²) and cube B (54 cm²), with volume of B (27 cm³), find the volume of cube A.

  7. Find xx if volume of smaller prism = 25 cm³ and volume of larger prism = 200 cm³.

Solutions for Practice Questions

  • Worked solutions can be found in the corresponding worksheet file.