Unlocking the Secrets of Similar Polygons: Shape, Size, and Proportions!

Step-by-step Guide: Similar Polygons

Defining Similar Polygons:
Two polygons are considered similar if they have the same shape but possibly different sizes. This implies two main conditions: their corresponding angles are congruent, and their corresponding sides are in proportion.

Criteria for Similarity:

• Angles: Every corresponding angle in one polygon must be congruent to its counterpart in the other polygon.
• Sides: The lengths of corresponding sides of the polygons must be in the same ratio.

Determining the Scale Factor:
The ratio of any two corresponding lengths in two similar geometric figures is called the ‘scale factor’. If one polygon is a scaled version of another, the amount by which it has been scaled is the scale factor. $\text{Scale Factor} = \frac{\text{Length of a side in the larger polygon}}{\text{Length of the corresponding side in the smaller polygon}}$

Examples

Example 1:
Triangles $ABC$ and $DEF$ have angles measuring $40^\circ$, $70^\circ$, and $70^\circ$ respectively for both. If $AB = 5 \text{ cm}$, $BC = 10 \text{ cm}$ and $DE = 2.5 \text{ cm}$, $EF = 5 \text{ cm}$, are they similar? If so, what’s the scale factor?

Solution:
Since all corresponding angles are congruent, they are potentially similar. To confirm, let’s check the side ratios:
$\frac{DE}{AB} = \frac{2.5 \text{ cm}}{5 \text{ cm}} = 0.5$ and $\frac{EF}{BC} = \frac{5 \text{ cm}}{10 \text{ cm}} = 0.5$
Since the ratios are equal, the triangles are similar with a scale factor of $0.5$.

Example 2:
Rectangles $WXYZ$ and $PQRS$ have widths of $4 \text{ cm}$ and $8 \text{ cm}$ and lengths of $6 \text{ cm}$ and $12 \text{ cm}$ respectively. Are they similar?

Solution:
For rectangles, the angles are all $90^\circ$, so only the side ratios need to be checked:
$\frac{PQ}{WX} = \frac{8 \text{ cm}}{4 \text{ cm}} = 2$ and $\frac{QR}{XY} = \frac{12 \text{ cm}}{6 \text{ cm}} = 2$
With equal ratios and congruent angles, the rectangles are similar.

Practice Questions:

1. Quadrilaterals $MNPQ$ and $RSTU$ have sides measuring $2 \text{ cm}$, $3 \text{ cm}$, $4 \text{ cm}$, $6 \text{ cm}$ and $4 \text{ cm}$, $6 \text{ cm}$, $8 \text{ cm}$, $12 \text{ cm}$ respectively. Are they similar?
2. Two triangles $GHI$ and $JKL$ have angles measuring $45^\circ$, $45^\circ$, and $90^\circ$ for both. If $GH = 7 \text{ cm}$, $HI = 10 \text{ cm}$ and $JK = 14 \text{ cm}$, $KL = 20 \text{ cm}$, are they similar?

Answers:

1. Yes, they are similar.
2. Yes, they are similar.