Congruence and Similarity

Ontario 10 Academic (MPM2D)

Transformations and similarity

Lesson

In Changing Shapes, we looked at how congruent shapes may be transformed in one or more ways on a number plane. We can also transform similar shapes. These similar shapes will be dilated by a scale factor (ie. enlarged or reduced by a certain ratio) in addition to the transformation. The video attached to the examples below explains this process.

Examples

Consider the figures shown.

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Are the two triangles congruent, similar or neither?

Congruent

ASimilar

BNeither

CCongruent

ASimilar

BNeither

CWhat is the transformation from triangle $ABC$

`A``B``C`to triangle $A'B'C'$`A`′`B`′`C`′?Dilation

AReflection

BRotation

CTranslation

DDilation

AReflection

BRotation

CTranslation

DWhat is the scale factor of the dilation from triangle $ABC$

`A``B``C`to triangle $A'B'C'$`A`′`B`′`C`′?

Consider the quadrilateral with vertices at $A$`A`$\left(-3,-3\right)$(−3,−3), $B$`B`$\left(-3,3\right)$(−3,3), $C$`C`$\left(3,3\right)$(3,3) and $D$`D`$\left(3,-3\right)$(3,−3), and the quadrilateral with vertices at $A'$`A`′$\left(-9,-9\right)$(−9,−9), $B'$`B`′$\left(-9,9\right)$(−9,9), $C'$`C`′$\left(9,9\right)$(9,9) and $D'$`D`′$\left(9,-9\right)$(9,−9).

Are the two rectangles similar, congruent or neither?

congruent

Asimilar

Bneither

Ccongruent

Asimilar

Bneither

CWhat is the transformation from rectangle $ABCD$

`A``B``C``D`to rectangle $A'B'C'D'$`A`′`B`′`C`′`D`′?dilation

Areflection

Brotation

Ctranslation

Ddilation

Areflection

Brotation

Ctranslation

DWhat is the scale factor of the dilation of rectangle $ABCD$

`A``B``C``D`to rectangle $A'B'C'D'$`A`′`B`′`C`′`D`′?

The quadrilateral with vertices at $\left(9,9\right)$(9,9), $\left(0,9\right)$(0,9), $\left(0,0\right)$(0,0) and $\left(9,0\right)$(9,0) is rotated 90 degrees clockwise around the origin and dilated by a factor of 2 with the origin as the centre of dilation.

What are the new coordinates of the vertices of the quadrilateral?

Write all four coordinates on the same line, separated by commas.

Verify, through investigation, the properties of similar triangles