9-4 Symmetry

Quick Review

A figure has reflectional symmetry or line symmetry if there is a reflection for which it is its own image.

A figure that has rotational symmetry is its own image for some rotation of 180 ° 180 degrees  or less.

A figure that has point symmetry has 180 ° 180 degrees  rotational symmetry.

Example

How many lines of symmetry does an equilateral triangle have?

An equilateral triangle reflects onto itself across each of its three medians. The triangle has three lines of symmetry.

Exercises

Tell what type(s) of symmetry each figure has. If it has line symmetry, sketch the figure and the line(s) of symmetry. If it has rotational symmetry, state the angle of rotation.

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19. How many lines of symmetry does an isosceles trapezoid have?
20. What type(s) of symmetry does a square have?
21. Give an example of a three-dimensional object that has rotational symmetry about a line.

9-5 Dilations

Quick Review

The diagram shows a dilation with center C and scale factor n. The preimage and image are similar.

In the coordinate plane, if the origin is the center of a dilation with scale factor n, then P ( x , y ) → P ′ ( n x , n y ) . p open x comma y close rightwards arrow , p prime , open n x comma n y close .

Example

The blue figure is a dilation image of the black figure. The center of dilation is A. Is the dilation an enlargement or a reduction? What is the scale factor?

The image is smaller than the preimage, so the dilation is a reduction. The scale factor is image length original length = 2 2 + 4 = 2 6 , or  1 3 . fraction imagelength , over originallength end fraction . equals . fraction 2 , over 2 plus 4 end fraction . equals , 2 sixths , comma or , 1 third , .

Exercises

22. The blue figure is a dilation image of the black figure. The center of dilation is O. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor.

    

Graph the polygon with the given vertices. Then graph its image for a dilation with center (0, 0) and the given scale factor.

23. M ( − 3 , 4 ) , A ( − 6 , − 1 ) , T ( 0 , 0 ) , H ( 3 , 2 ) ; m open negative 3 comma 4 close comma eh open negative 6 comma negative 1 close comma t open 0 comma 0 close comma h open 3 comma 2 close semicolon  scale factor 5
24. F ( − 4 , 0 ) , U ( 5 , 0 ) , N ( − 2 , − 5 ) ; f open negative 4 comma 0 close comma u open 5 comma 0 close comma n open negative 2 comma negative 5 close semicolon  scale factor 1 2 1 half
25. A dilation maps Δ L M N cap delta l m n  onto Δ L ′ M ′ N ′ . L M = 36  ft , L N = 26  ft , M N = 45  ft, cap delta , l prime , m prime , n prime , . l m equals 36 , ft comma l n equals 26 , ft comma m n equals 45 , ftcomma  and L ′ M ′ = 9  ft . l prime , m prime , equals 9 ft .  Find L ′ N ′ l prime , n prime  and M ′ N ′ . m prime , n prime , .

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