MYP Extended Mathematics Topic Similar Shapes and Dilation Revision Notes

MYP Extended Mathematics Topic Similar Shapes and Dilation Revision Notes | RevisionDojo

What Does “Similar” Mean?

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

Definition

Similar figures

Two figures are similar when (1) their corresponding angles are equal, and (2) all the corresponding side lengths are in the same ratio (they are all multiplied by the same scale factor).

We write $A∼B$ to mean “figure $A$ is similar to figure $B$”.
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

Definition

Dilation

Also called an enlargement or a reduction, is a transformation that changes the size of a figure but keeps its shape.

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

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

Example

If a rectangle is dilated with scale factor $r=3$, all side lengths are multiplied by 3.
If a triangle is dilated with scale factor $r=0.4$, all side lengths become 40% of the original.

Center of Dilation

Sometimes a dilation just uses a scale factor (for example, scaling a drawing on a photocopier).
In geometry, we often also specify a center of dilation.
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 $P$.
Symbolically, if $C$ is fixed and $r\neq 0$,
1. when $r>0$, $P′$ lies on the same side of $C$ as $P$,

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MYP Extended Mathematics Similar Shapes and Dilation Notes

What Does “Similar” Mean?

Dilation and Scale Factor

Center of Dilation

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What Does “Similar” Mean?

Dilation and Scale Factor

Center of Dilation

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Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

MYP Extended Mathematics Similar Shapes and Dilation Notes

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

What Does “Similar” Mean?

Dilation and Scale Factor

Center of Dilation

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

What Does “Similar” Mean?

Dilation and Scale Factor

Center of Dilation

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account