LESSON 6 SIMILARITY AND ITS APPLICATIONS

LESSON 6 SIMILARITY AND ITS APPLICATIONS

“Mathematics is the art of giving the same name to different things.” Poincaré

1. SIMILAR SHAPES Similar figures are identical in shape, but not in size.

For example, two circles are always similar.

Two squares are always similar:

And two rectangles could be similar:

But will probably not be. In real life there are plenty of similar shapes: photographs, movies, maps, etc. The scale of a map is the relationship between a distance in the map and the corresponding distance in ground. A scale of 1 : 100 000 means that the real distance is 100 000 times the length of 1 unit on the map or drawing.

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Example: Write the scale 1 cm to 1 m in ratio form.

Solution:

Areas and volumes If the ratio of two similar shapes is k, the ratio of their areas is k2 ant the ratio of their volumes is k3 .

2. RECTANGLES THAT HAVE INTERESTING PROPORTIONS DIN A-4 The dimensions of the paper we use in photocopiers and printers have an interesting denomination: A4, a format that is used throughout the world. A4 pages have a width of 210mm and a length of 297mm; an odd measure, since it would seem to make more sense to use round numbers. Why not 20cm by 30cm, for example? Usual pieces of paper (DIN A-4) have a curious propierty: if you cut it in half, every on of the pieces is similar to the previous one. l l/2

1 1

l = 1 As these two rectangles are similar, their sides are in proportion: 1 l . 2 l1 l 2 = → =1 →l2 = 2 → l = 2 1l 2 2

Thus, if a rectangle is similar to its half, the proportion between its sides is 2 . The sides of the rectangle must be in the proportion of 1 to 2 (approximately 1.4142). Do the calculation for yourself and you will see that 210 x 1.4142 = 296.982; practically 297. We have found the proportions of A4. There are a series of basic formats of paper that start at A0, (the largest), followed by A1, A2 , A3 and so forth down to the very small A10; 26mm by 37mm. All these formats

2/8 IES UNIVERSIDAD LABORAL Mercedes López LESSON 6 SIMILARITY AND ITS APPLICATIONS are obtained by folding a page in half in order to obtain the format above. This means that, for example, by folding A0 in half we obtain A1, and that by folding A4, we obtain A5.

GOLDEN RATIO

The rectangle above has a peculiarity: if you remove a square from it, the rectangle that remains is similar to the original.

Thus, Φ 1 = → Φ2 − Φ −1 = 0 1Φ − 1 and solving the quadratic equation, you get: + Φ = 1 5 2 That is the golden ratio that you studied in lesson 1 as an irrational number. To construct a golden rectangle, draw a square. Draw a line from the midpoint of one 5 side of the square to an opposite corner( AB = ). Use that line as the radius to draw 2 an arc that defines the height of the rectangle. Complete the golden rectangle.

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3. SIMILAR TRIANGLES Thales intercept theorem If the lines a, b and c are parallels and intersect another two parallels lines r and s, the segments determined by them are proportional.

Similar triangles Two triangles are similar if: 1. The three sides are in the same proportion.

2. The three angles of the first triangle are equal to the three angles of the second triangle.

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The triangles ABC and ADE are similar and we say they are in 'Thales position'.

Rules for proving triangles similar AA rule: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

SSS rule: If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.

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SAS rule: If an angle of one triangle is equal to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar.

Copy and solve the following problems 1.

4. SIMILARITY IN RIGHT TRIANGLES

Two right triangles are SIMILAR if one of the acute angles of the first is equal to one of the acute angles of the second. This conclusion is supported by the following reasons: 1. The right angle in the first triangle is equal to the right angle in the second, since all right angles are equal. 2. The sum of the angles of any triangle is 180°. Therefore, the sum of the two acute angles in a right triangle is 90°. 3. Let the equal acute angles in the two triangles be represented by A and A’ respectively. Then the other acute angles, B and B’, are as follows: B = 90° - A B’ = 90° - A’

1 http://www.regentsprep.org/Regents/math/geometry/GP11/PracSim.htm

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4. Since angles A and A’ are equal, angles B and B’ are also equal, we conclude that two right triangles with one acute angle of the first equal to one acute angle of the second have all of their corresponding angles equal. Thus the two triangles are similar. Theorem 1: In all right triangles, the length one of the legs is a mean proportional between the hypotenuse and the projection on it.

Theorem 2: In all right triangles, the height of the hypotenuse is a mean proportional between the two segments that it divides.

Examples:

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1. The hypotenuse of a right triangle measures 30 cm and the projection of a leg is 10.8 cm on the hypotenuse. Find the length of the other leg.

2. In a right triangle, the projections of the legs on the hypotenuse are 4 and 9 cm in length. Calculate the height of the triangle.

Copy and solve the following exercises 2.

2 http://www.regentsprep.org/Regents/math/geometry/GP12/PracMeanP.htm

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