7.5.A 7.5.C

Similarity 7th Grade

Texas Essential Knowledge and Skills (TEKS): 7.5.A
generalize the critical attributes of similarity, including ratios within and between similar shapes;
Texas Essential Knowledge and Skills (TEKS): 7.5.C
solve mathematical and real-world problems involving similar shape and scale drawings.

Similar figures are the same shape, but not necessarily the same size.

Scale factor is the ratio of corresponding side lengths
(new image to old image) of similar figures.

\begin{array}{rcl} scale factor & = & \frac{new length}{old length} \\ = & \frac{12 ft}{4 ft} \\ = & 3 \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.

A scale drawing (or scale model) is a proportional representation of an object.

A scale drawing's scale indicates the relationship between the
representation's measurements and the actual measurements.

\begin{array}{rcl} \frac{1 in.}{14.4 ft} & = & \frac{5 in.}{x} \\ 1 in. (x) & = & 5 in. (14.4 ft) \\ 1 in. (x) & = & 72 in. ft \\ 1 in. (x) \div 1 in. & = & 72 in. ft \div 1 in. \\ x & = & 72 ft \end{array}

Area	=	base × height
	=	$\frac{5}{8}$ in. × $1 \frac{1}{4}$ in.
	=	$\frac{5}{8}$ in. × $\frac{5}{4}$ in.
	=	$\frac{25}{32}$ sq in.

The scale for area is the scale for length squared.
So, the scale for area is 1 square inch to (14.4)^² square feet, which is 1 square inch to 207.36 square feet.
To find the area in square feet, multiply

\frac{25}{32}

square inch by 207.36 sq ft/sq in.

207.36 sq ft/sq in. ×

\frac{25}{32}

sq in. = 162 sq ft

Similarity

Texas Essential Knowledge and Skills (TEKS): 7.5.A

generalize the critical attributes of similarity, including ratios within and between similar shapes;

Texas Essential Knowledge and Skills (TEKS): 7.5.C

solve mathematical and real-world problems involving similar shape and scale drawings.

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