Understanding Similar Plane Objects
The Mathematical Rules of Similarity
Similarity is a specific relationship between shapes. For two polygons (shapes with straight sides, like triangles, rectangles, or pentagons) to be similar, two main conditions must be met:
- Corresponding Angles are Congruent: This means each angle in one shape is exactly equal in measure to its matching angle in the other shape. If shape A has angles of 40°, 60°, and 80°, then shape B must also have angles of 40°, 60°, and 80° in the corresponding positions.
- Corresponding Sides are Proportional: The lengths of the sides are not equal, but they are all multiplied (or divided) by the same number. This number is the scale factor or similarity ratio.
These two conditions are interconnected. For triangles, a special rule exists: if the angles of one triangle are congruent to the angles of another triangle, then the triangles are automatically similar, and their sides are proportional. This is called Angle-Angle-Angle (AAA) similarity. For other polygons, like quadrilaterals, having equal angles is not enough; the sides must also be proportional.
$ k = \frac{\text{Length of a side in the new shape}}{\text{Length of the corresponding side in the original shape}} $
If $k > 1$, the new shape is an enlargement. If $0 < k < 1$, it is a reduction.
Consider two similar rectangles. Rectangle P has sides of 3 cm and 5 cm. Rectangle Q is larger, with corresponding sides of 6 cm and 10 cm. First, we check the angles: all angles in both rectangles are 90°, so they are congruent. Next, we check the sides. The ratio for the shorter sides is 6 / 3 = 2. The ratio for the longer sides is 10 / 5 = 2. Since the ratio is the same (2), the sides are proportional. Therefore, the rectangles are similar with a scale factor of 2.
| Original Object | Similar Object | Scale Factor | Why They Are Similar |
|---|---|---|---|
| A standard 3" x 5" index card | A giant poster sized 6 ft x 10 ft | 24 (Converted to same units) | All angles are 90°; side ratios are equal (6ft/3in = 10ft/5in = 24). |
| A triangle with sides 3, 4, 5 | A triangle with sides 6, 8, 10 | 2 | By the SSS (Side-Side-Side) similarity rule: all side ratios are equal (6/3 = 8/4 = 10/5 = 2). |
| A circle of radius 2 cm | A circle of radius 5 cm | 2.5 | All circles are similar! Their shapes are identical, only size differs. The scale factor is the ratio of any corresponding measurements (radii, diameters, circumferences). |
Similar Triangles: The Special Case
Triangles are the simplest polygons and have the most powerful similarity rules. Because a triangle's shape is completely determined by its angles (try to draw two different triangles with the same three angles—you can't!), the rules for triangle similarity are easier to satisfy than for other shapes.
The main similarity postulates for triangles are:
- AA (Angle-Angle): If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. (Since the sum of angles in a triangle is always 180°, the third angles must also be equal).
- SSS (Side-Side-Side): If the ratios of all three corresponding sides of two triangles are equal, the triangles are similar.
- SAS (Side-Angle-Side): If the ratio of two corresponding sides is equal and the angle between those sides is congruent, the triangles are similar.
Imagine a flagpole casting a shadow. You can measure the length of the shadow. You can also place a meter stick upright and measure the length of its shadow. The sun's rays create the same angle for both the flagpole and the meter stick. This forms two triangles that share the same two angles (the right angle where the objects meet the ground, and the angle of the sun's rays). By the AA postulate, the triangles are similar. Therefore, you can set up a proportion to find the height of the flagpole:
$ \frac{\text{Height of flagpole}}{\text{Length of its shadow}} = \frac{\text{Height of meter stick}}{\text{Length of its shadow}} $
This is a classic and practical application of similar triangles in real-world measurement.
Maps, Models, and Scale Drawings
One of the most common real-world uses of similarity is in creating maps, scale models, and architectural blueprints. These are all similar representations of a larger (or sometimes smaller) object.
A scale on a map, like 1 : 50,000, is the scale factor. It tells us that 1 unit of distance on the map (e.g., 1 cm) represents 50,000 of the same units in reality (50,000 cm, or 0.5 km). Every feature on the map—roads, rivers, parks—is a similar, reduced version of the actual landscape. The angles of turns in a road are preserved, and the relative distances are kept proportional.
Similarly, an architect's blueprint of a house is similar to the actual house. If the living room on the blueprint is a rectangle measuring 10 cm x 15 cm with a scale of 1 : 100, then the actual living room will be 10 m x 15 m. The shape, the angles of the walls, and the placement of windows are all identical, just scaled up.
Solving Problems with Similar Figures
When faced with a problem involving similar figures, we use the proportionality of their sides. This is often done by setting up and solving a proportion equation.
Example Problem: Triangle ABC is similar to triangle DEF. Side AB = 8 cm, BC = 12 cm, and AC = 10 cm. The corresponding side to AB in triangle DEF (side DE) is 12 cm. Find the lengths of the other two sides of triangle DEF.
Step 1: Find the scale factor. The scale factor from triangle ABC to triangle DEF is the ratio of corresponding sides: $ k = \frac{DE}{AB} = \frac{12}{8} = 1.5 $.
Step 2: Apply the scale factor to the other sides.
- Side EF corresponds to BC: $ EF = BC \times k = 12 \times 1.5 = 18 $ cm.
- Side DF corresponds to AC: $ DF = AC \times k = 10 \times 1.5 = 15 $ cm.
We can also organize this information in a table for clarity:
| Corresponding Sides | Triangle ABC (Original) | Scale Factor (k = 1.5) | Triangle DEF (Similar) |
|---|---|---|---|
| AB and DE | 8 cm | $\times 1.5$ | 12 cm |
| BC and EF | 12 cm | $\times 1.5$ | 18 cm |
| AC and DF | 10 cm | $\times 1.5$ | 15 cm |
Important Questions
Q1: Are all squares similar? What about all rectangles?
Answer: Yes, all squares are similar. A square is defined as a quadrilateral with four equal sides and four right angles. Any two squares meet the similarity conditions: all corresponding angles are 90°, and the ratio of any two corresponding sides is constant (the ratio of their side lengths). For example, a square of side 2 cm and a square of side 5 cm are similar with a scale factor of 2.5.
However, not all rectangles are similar. While all rectangles have four 90° angles, their sides are only proportional if the ratio of length to width is the same. A 2x4 rectangle and a 3x6 rectangle are similar (ratio 2/3 = 4/6). But a 2x4 rectangle and a 3x5 rectangle are not similar (2/3 ≠ 4/5).
Q2: What is the difference between similar and congruent shapes?
Answer: Congruent shapes are identical in both shape and size. They are essentially the same figure, just possibly rotated or flipped. You could place one perfectly on top of the other. Similar shapes are identical in shape only, not necessarily in size. All congruent shapes are similar (with a scale factor of 1), but not all similar shapes are congruent.
Think of it this way: Two photographs printed from the same negative are congruent if they are the same size. They are similar if one is a wallet-sized photo and the other is a poster.
Q3: How can I prove two triangles are similar without knowing all the side lengths?
Answer: Use the Angle-Angle (AA) similarity postulate. This is often the simplest method. If you can show that two angles of one triangle are equal in measure to two angles of the other triangle, the triangles are similar. You don't need any side lengths at all for this proof. This works because the sum of angles in a triangle is fixed at 180°, so if two angles match, the third automatically matches as well.
For example, if you have a triangle where you know two angles are 50° and 70°, and another triangle where two angles are 50° and 70°, you have proven they are similar, regardless of how big or small they are.
Footnote
1 AAA (Angle-Angle-Angle): A postulate/theorem in geometry stating that if the three angles of one triangle are congruent to the three angles of another triangle, then the two triangles are similar.
2 SSS (Side-Side-Side) Similarity: A rule stating that if the ratios of all three pairs of corresponding sides of two triangles are equal, then the triangles are similar.
3 SAS (Side-Angle-Side) Similarity: A rule stating that if the ratio of two pairs of corresponding sides is equal and the included angles are congruent, then the triangles are similar.
4 Scale Factor: The constant ratio of any two corresponding lengths in two similar geometric figures.
5 Corresponding Sides/Angles: Sides or angles that are in the same relative position in two similar or congruent geometric figures.