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.