Download Similarity: Key Terms Term Definition Example Transformation A

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