8.3.a 8.3.b 8.3.c MA.8.2.2 MA.8.2.4 8.10.d

Dilations 8th Grade

Texas Essential Knowledge and Skills (TEKS): 8.3.a
generalize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dilation;
Texas Essential Knowledge and Skills (TEKS): 8.3.b
compare and contrast the attributes of a shape and its dilation(s) on a coordinate plane; and
Texas Essential Knowledge and Skills (TEKS): 8.3.c
use an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation.
Texas Essential Knowledge and Skills (TEKS): 8.10.d
model the effect on linear and area measurements of dilated two-dimensional shapes.
Florida - Benchmarks for Excellent Student Thinking: MA.8.2.2
Given a preimage and image generated by a single dilation, identify the scale factor that describes the relationship.
Florida - Benchmarks for Excellent Student Thinking: MA.8.2.4
Solve mathematical and real-world problems involving proportional relationships between similar triangles.

A dilation is a transformation in which a figure's size changes.
The image which results from a dilation is similar to the original image.
Corresponding sides of similar figures are proportional.

Parallelogram EFGH is a dilation of parallelogram ABCD.
So, parallelograms ABCD and EFGH are similar.
This means that the ratios of the corresponding sides are all equal.

\frac{AB}{EF} = \frac{BC}{FG} = \frac{CD}{GH} = \frac{DA}{HE}

\begin{array}{rcl} \frac{AB}{EF} & = & \frac{BC}{FG} \\ \frac{5.1 cm}{EF} & = & \frac{3.6 cm}{1.8 cm} \\ (5.1 cm)(1.8 cm) & = & (EF)(3.6 cm) \\ \frac{(5.1 cm)(1.8 cm)}{(3.6 cm)} & = & EF \\ 2.55 cm & = & EF \end{array}

If the scale factor from figure A to figure B is k,
then the area of figure B is k^² times the area of figure A.

Figure 1 has side lengths: a, b, and c.
A scale factor of k has been applied to figure 1 to produce figure 2.
So, figure 2 has side lengths: k times a, k times b, and k times c.

The area of a triangle is one-half times the base times the height.
Examine the areas of both figures.

Area_{_{figure 1}}	=	$\frac{1}{2}$ ab
Area_{_{figure 2}}	=	$\frac{1}{2}$ (ka)(kb)
	=	kk( $\frac{1}{2}$ ab)
	=	k^²(Area_{_{figure 1}})

So, the area of figure 2 is k^² times the area of figure 1.

Dilations

Texas Essential Knowledge and Skills (TEKS): 8.3.a

generalize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dilation;

Texas Essential Knowledge and Skills (TEKS): 8.3.b

compare and contrast the attributes of a shape and its dilation(s) on a coordinate plane; and

Texas Essential Knowledge and Skills (TEKS): 8.3.c

use an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation.

Texas Essential Knowledge and Skills (TEKS): 8.10.d

model the effect on linear and area measurements of dilated two-dimensional shapes.

Florida - Benchmarks for Excellent Student Thinking: MA.8.2.2

Given a preimage and image generated by a single dilation, identify the scale factor that describes the relationship.

Florida - Benchmarks for Excellent Student Thinking: MA.8.2.4

Solve mathematical and real-world problems involving proportional relationships between similar triangles.

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