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12-1 Size Transformations RevisitedPowerPoint Presentation

12-1 Size Transformations Revisited

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12-1 Size Transformations Revisited

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12-1 Size Transformations Revisited

Size transformations can be accomplished with any point as center and do not require coordinate geometry.

Seven dolls are pictured on page 717. The widths of their faces, measured across the widest part, are shown in table below. If the face of the sixth-largest doll (front row second from the right) is viewed as having a width of 1 unit, what are the widths of the faces of the other 6 dolls? Round the answers to the nearest thousandth.

Definitions of Size Change (Size Transformation) with Any Center, Magnitude, and Size-Change Factor

Let O be a point and k be any nonzero real number. For any point P, let S(P) = P’ be the point on lineOP with OP’ = k · OP in the direction of rayOP if k is positive and in the direction opposite rayOP if k is negative. Then S is the size change or size transformation with center O and magnitude or size-change factor k.

Theorem

The transformation Skthat maps (x, y) onto (kx, ky), with k ¹ 0, is a size transformation S with center (0, 0) and magnitude k.

Size-Change Preservation Properties Theorem

Every size transformation preserves:

- angle measure
- betweenness
- collinearity.

Suppose S is a size transformation of magnitude 0.45. If DMNP has a perimeter of 84 centimeters, find the perimeter of S(DMNP).

Find the lengths of the sides and the area of DPQR with vertices P = (1, 1), Q = (8, 1), and R = (8, 5).

Now find the lengths of the sides and the area of DXYZ if the transformation S3, with center (0, 0), maps DPQR onto DXYZ.

Figure Size-Change Theorem

If a figure is determined by certain points, then its size-change image is the corresponding figure determined by the size-change images of those points.

Let’s explore the transformation Sm, nthat maps (x, y) onto (mx, ny).

- a. What is the slope of the image of the line y = x under Sm, n?
- b. If a figure has area A, what is the area of the image of that figure if each point is transformed by Sm, n?
- c. What is an expression for the length of the segment whose endpoints are (a, b) and Sm, n(a, b)?