# In a picture formed as a diamond, show the results of each type of transformation. a. Dilation by a factor of two (that is, double the length of each line segment) b. Rotation by 90° clockwise...

In a picture formed as a diamond, show the results of each type of transformation. a. Dilation by a factor of two (that is, double the length of each line segment)

b. Rotation by 90° clockwise (that is, a clockwise quarter turn) c. Translation up by some distance x d. Reflection across a horizontal line of reflection

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### 1 Answer

On a coordinate system label the points A(0,2) B(1,0) C(0,-2) and D(-1,0) and connect them to form a diamond shape.

(1) A dilation factor 2` `` `: The instruction to double each side length is ambiguous (You could apply a vertical or horizontal stretch of factor 2and double the side length, but I don't think that is what is requested)

In the drawing, the length of AB is `sqrt(5)` (From the Pythagorean theorem: `1^2+2^2=(AB)^2==>AB=sqrt(1+4)` )

If we move A to A'(0,4) and B ro B'(2,0) the length of A'B' is `sqrt(4^2+2^2)=sqrt(20)=2sqrt(5)` which is twice as long as AB. Then move C to C'(0,-4) and D to D'(-2,0)

We performed the transformation `(x,y)->(2x,2y)`

(2) Rotation `90^@` clockwise: A moves to A'(2,0) B to B'(0,-1) C to C'(-2,0) and D to D'(0,1)

` `We performed the transformation `(x,y)=>(y,-x)`

(3) Translation up c; let c=3 Then A goes to A'(0,5) B to B'(1,3) C to C'(0,1) and D to D'(-1,3)

The transformation `(x,y)->(x,y+3)`

(4) Reflection over a horizontal axis; let the axis be y=-2

Then A goes to A'(0,-6) B to B'(1,-4) C to C'(0,-2) (It is on the line of reflection) and D to D'(-1,-4)

The transformation `(x,y)=>(x,-y-4)`

The graphs: original in black, dilation in red, rotation in green, translation in blue, reflection in purple