**Similarity and Congruence**

Two objects are similar if they have the same shape, so that one is an enlargement of the other.

Two objects are congruent if they are the same shape and size.

Corresponding angles are equal. Corresponding lengths are in the same ratio – they have the same Scale Factor. The scale factor is greater than 1 for an enlargement. The scale factor is smaller than 1 for a reduction.

**Example**

3:2 or 1.5 or 3/2

Two shapes are said to be similar if one is a scaled version of the other. This means that their corresponding angles are equal; their corresponding sides are in the same ratio. (ie. they have the same scale factor) similar?

Check that their corresponding angles are equal. Check the ratio of their corresponding sides.

Length of A/Length of B

= 20 mm/30 mm

= 2/3

Breadth of A/Breadth of B

10 mm/15 mm

2/3

x = scale factor x corresponding side

Scale factor = 5/20 = ¼

x = scale factor x corresponding side

x = ¼ x 12

x = 3

For any triangle

Split the triangles into their components

x = scale factor X corresponding side

scale factor = AB/AD = 12/15 = 4/5

corresponding side = DE = 10 cm

x = 4/5 X 10

x = 40/5 = 8 cm

x = scale factor X corresponding side

Area A = 6 cm^{2 }

Scale factor = 2

(Scale factor)^{2 } = 4

Area B = 24 cm^{2 }

Area B = 4 X 6

= 4 X Area A

Area B = (Scale factor)^{2 } X original area

Scaled area = (Scale factor)^{2 } X original area

Area B = (Scale factor)^{2 } X original area

= 2^{2 X 16}

^{ } = 4 X 16

= 64 cm^{2}

Area A = 6cm^{2 }

Volume A = 6 cm^{2 }

(Scale factor)^{2 } = 8

Area B = 24 cm^{2 }

Volume B = 48cm^{3 }

Volume B = 8 X 6 = 8 X Volume A

Scale Volume = (Scale factor)^{3 }x original volume

New Volume = (scale factor)^{3 }X original volume

= (1/3)^{3 }X 200

= 1/8 X 200

= 1/8 X 200

= 25 ml

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