Similar shapes and dilation Notes

What Does “Similar” Mean?

1. You already know what it means for two photos to be the “same picture, just resized”.
2. In geometry, we call such figures similar.

1. We write $A∼B$ to mean “figure $A$ is similar to figure $B$”.
2. If one shape is similar to another, it is a scaled copy: the shape is the same, only the size changes (and possibly the position or orientation).

Dilation and Scale Factor

A dilation is determined by a scale factor $r$, which is the ratio of corresponding sides of the original figure and its image.

1. If $|r|>1$, the image is an enlargement (bigger).
2. If $0 < |r| <1$, the image is a reduction (smaller).
3. $|r|$ tells you how many times larger or smaller the image is compared to the original.

Center of Dilation

1. Sometimes a dilation just uses a scale factor (for example, scaling a drawing on a photocopier).
2. In geometry, we often also specify a center of dilation.
3. In a dilation with scale factor $r$ and center of dilation $C(x,y)$:
   1. Each point $P$ in the original is mapped to a point $P′$ on the line $CP$.
   2. The distance from $C$ to $P′$ is $|r|$ times the distance from $C$ to $P$.
4. Symbolically, if $C$ is fixed and $r\neq 0$,
   1. when $r>0$, $P′$ lies on the same side of $C$ as $P$,
   2. when $r<0$, $P′$ lies on the opposite side of $C$.

Coordinate Rules for Dilation

Dilation from the Origin

If the center of dilation is the origin $(0,0)$ and the scale factor is $r$, then $$(x, y) \longmapsto(r x, r y)$$

Dilation from a Point $(a,b)$

1. If the center of dilation is $C(a,b)$ and the scale factor is $r$, $$(x, y) \longmapsto(a+r(x-a), b+r(y-b))$$
2. This moves the point relative to the chosen center, not just the origin.

Similarity and Dilation

Every dilation produces a figure similar to the original:

1. Corresponding angles stay equal.
2. Corresponding side lengths are multiplied by rr.
3. So the ratio $$\frac{\text { image side }}{\text { original side }}=|r|$$ s the same for all corresponding sides.

How Size and Area Change

If a figure is dilated by a scale factor $r$:

1. Every length (sides, diagonals, perimeter) is multiplied by $|r|$.
2. Every area is multiplied by $|r|^2$.
3. In 3D, every volume would be multiplied by $|r|^3$.

Recognizing Similar Figures

1. To decide whether two figures are similar, check:
   1. Angles – all corresponding angles are equal.
   2. Side ratios – all corresponding side lengths are in the same ratio (the same scale factor).
2. For triangles, this can be simplified using familiar tests:
   1. AA (angle–angle): two angles equal → triangles are similar.
   2. SSS proportional: three pairs of sides in the same ratio → similar.
   3. SAS proportional: two pairs of sides in the same ratio and the included angle equal → similar.

Positive vs Negative Scale Factors

When $r>0$

1. The image is on the same side of the center as the original.
2. If $|r|>1$: enlargement
3. If $0<|r|<1$: reduction
4. Orientation (clockwise vs anticlockwise) is preserved.

When $r<0$

1. The image lies on the opposite side of the center.
2. The figure is still similar (same shape, scaled by $|r|$), but it has also been rotated 180° about the center.
3. The change in sign affects position, not the size: $|r|$ still controls how big it is.