Geometry Questions: Rotation and Dilation?

1.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 ?

2.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?

3.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?

4.What values of the scale factor create an enlargement? What values create a reduction?

5.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?

6.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’?

7.A triangle has coordinates A (2, 5), B (-2, 8) and C (1, -4). What are the new coordinates if the triangle is dilated with a scale factor of 4 ?

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

*Thanks SOOO much in advance; if your a student; i can return the help in english; science or history

1 Answer

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  • 9 years ago
    Best Answer

    Rotation:

    This is defined as movement of a point P(x,y) in a plane by an angle θ to another position Q(x₁,y₁), without changing its distance from the origin. Thus the point moves on a circle of radius r. In general the angle of movement is considered to be only in counter clockwise direction. If it is rotated through an angle of θ in clockwise direction, then the equivalent angle in counter clockwise direction is given by (360-θ).

    In these rotation in counter clockwise direction, the new position of Q(x₁,y₁), is given by:

    (x₁,y₁) = (xcosθ-ysinθ, xsinθ+ycosθ)

    Dilation:

    This is defined as either enlargement of reduction of a figure in plane movements. If the scale factor is greater than 1, 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 Q(x₁,y₁), is given by:

    (x₁,y₁) = (kx, ky), where 'k' is the scale factor.

    Solutions to the other questions asked:

    1) Scale factor is not given; however you can do yourself with the above forumula.

    2) Here it is Rotation by 90 deg. So, θ = 90;

    ==> cos90 = 0 and sin90 = 1;

    Thus, (x₁,y₁) = (x*0-y*1, x*1+y*0) = (-y,x)

    3) Here θ = 270; so, cos(270) = 0 and sin(270) = -1

    Thus, (x₁,y₁) = (x*0-y*-1, x*-1+y*0) = (y,-x) [It is equivalent to 90 deg rotation in clockwise]

    4) Already explained above.

    5) Here θ = 180 deg; cos(180) = -1 and sin(180) = 0

    So, (x₁,y₁) = (x*-1-y*0, x*0+y*-1) = (-x,-y)

    6) Here (3,6) moves to (2,4); thus 3k = 2; ==> scale factor k = 2/3, which well satisfies all other figures. So here the scale factor is 2/3, which implies the dilation causes reduction.

    The length of new line = 15 x 2/3 = 10 units.

    7&8) You can solve yourself using the above explained mathematical form.

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