Rotations and Dilations help?

Question

1.)How are the coordinates of the new point found if it is rotated 90° counterclockwise? How many degrees is that equivalent to if the rotation is clockwise?

 

2)How are the coordinates of the new point found if it is rotated 180° counterclockwise? How many degrees is that equivalent to if the rotation is clockwise?

 

3)Describe in your own words what a rotation is.

4)A triangle has coordinates A (1, 5), B (-2, 1) and C (0, -4). What are the new coordinates if the triangle is dilated with a scale factor of  1/5<!-- RspH:I -->? 

5)A line segment has endpoints P (3, 6) and Q (12, 18) and is dilated so that its new endpoints are P’ (2, 4) and Q’ (8, 12). What is the scale factor? If the length of PQ is 15, what is the length of P’Q’?

6)A triangle has coordinates A (1, 5), B (-2, 1) and C (0, -4). What are the new coordinates if the triangle is rotated 90° clockwise around the origin?

7)How are the coordinates of the new point found if it is rotated 270° counterclockwise? How many degrees is that equivalent to if the rotation is clockwise

8)How are the coordinates of the new point found if it is dilated with a scale factor of 3?  

9)Describe in your own words what a dilation is.

 

Answer 1

(1)

Let the point be .

The complete rotation is .

The point is rotated counter clock wise.

Rotation of axes:

When an equation of a conic section is rewritten in the -plane by rotating the coordinate axes through , the equation in xy - plane can be found using

and .

The new coordinates after rotation are ,

.

.

The new coordinates after rotation of  counter clock wise is .

The rotation is equal in clock wise direction is .

Answer 2

(2)

Let the point be .

The complete rotation is .

The point is rotated counter clockwise.

Rotation of axes:

When an equation of a conic section is rewritten in the -plane by rotating the coordinate axes through , the equation in xy - plane can be found using

and .

The new coordinates after rotation are ,

.

.

The new coordinates after rotation of  counter clockwise is .

The rotation is equal in clockwise direction is .

Answer 3

Rotation:

The "rotation" transformation is where you turn a figure about a given point .

The point about which the object is rotated can be inside the figure or anywhere outside it.

The amount of rotation is called the angle of rotation and is measured in degrees.

By convention a rotation counter-clockwise is a positive angle, and clockwise is considered a negative angle. 

Answer 4

 

 4)

Coordinates of triangle are and  .

Triangle is dilated with a scale factor of  .

Dilation:

This is defined as either enlargement of reduction of a figure in plane movements.

If the scale factor is greater than , then it is said to be enlargement.

If the scale factor is between 0 and 1, then it is said to be reduction.

In either case of Dilation, the new point is given by .

Thus, new coordinates of the triangle are

.

.

. 

 

Answer 5

(5)

The end points of the line segment are and .

The new end points are and .

Scale factor:

If the scale factor is then the new point  is .

Find the scale factor.

.

The scale factor is .

The length of the is .

The length of the is .

The length of the is .

Answer 6

(6)

A triangle has coordinates A (1, 5), B (-2, 1) and C (0, -4).

The point is rotated  clockwise around the origin.

The rotation is equal in counter clockwise direction is .

When an equation of a conic section is rewritten in the plane by rotating the coordinate axes through , the equation in -plane can be found using

 and .

The new coordinates after rotation are ,

.

.

The new coordinates after rotated  clockwise is .

Thus, new coordinates of the triangle are ,  and .

Answer 7

(7)

Let the point be .

The complete rotation is .

The point is rotated counter clockwise.

Rotation of axes:

When an equation of a conic section is rewritten in the -plane by rotating the coordinate axes through , the equation in xy - plane can be found using

and .

The new coordinates after rotation are ,

.

.

The new coordinates after rotation of  counter clockwise are .

The rotation in clockwise direction is .

Answer 8

(8)

Let the point be .

Point is dilated with a scale factor of .

Dilation:

If the coordinates of a point are dilated with a scale factor of then new point after dilation is .

New coordinates of the point are .

Answer 9

(9)

Dilation:

It is defined as either enlargement of reduction of a figure in plane movements.

If the scale factor is greater than , then it is said to be enlargement.

If the scale factor is between 0 and 1, then it is said to be reduction.