LAMC Junior Circle April 15, Constructing Triangles.

Transcription

1 LAMC Junior Cirle April 15, 2012 Olga Radko Oleg Gleizer Construting Triangles. Copyright: for home use only. This handout is a part of the book in preparation. Using it in any way other than the above is not permitted by the authors. The word polygon means multi-angled in the language of Eulid, the anient Greek. The simplest possible polygon in the Eulidean plane is a triangle, a polygon with three verties, sides, and angles. Let us learn various ways to onstrut a triangle using a ompass and a ruler. 1

2 Example 1 Using a ompass and a ruler, draw a triangle with the given sides a, b, and. a b Step 1. Draw a straight line and mark a point on it. Step 2. Measure side with a ompass, plae the needle at the marked point on the line, and mark side on the line. We will use straight parenthesis to denote the length of a straight line segment. For example, the length of the side a is a. Step 3. Now it is time to reall the definition of a irumferene. The points of the plane having the distane a from the left end of side form a irumferene of radius a entered at the left node of the above piture. Similarly, the points having the distane b from the right end of belong to the irumferene of radius b entered at the right node. 2

3 a b Step 4. The irumferenes interset at two points. Eah of them has the distane a from the left end of the side and the distane b from the right. We an pik either one as the third vertex of the triangle. a b 3

4 Problem 1 Use a ompass and a ruler to onstrut a triangle having the following sides. a b 4

5 Example 2 Using a ompass and a ruler, draw a given angle having a given ray as its side. Below, you will see two different solutions to this problem, eah having its own merit. Solution 1. The following approah is widely used in Mathematis. We need to solve a problem, but we don t know how. Instead, let us solve a problem that we know how to solve. In this partiular ase, we already know how to onstrut a triangle with given side lengths. Instead of solving the original problem, let s do that! The word auxiliary means providing supplementary or additional help and support as in an auxiliary nanny, a nanny oasionally employed in addition to the main one. As many more words used in siene, this one originates from Latin, the language of the anient Rome. Its progenitor, the Latin word auxilium means help. 5

6 Step 1. Let us draw an auxiliary triangle having as its angle. Traditionally, the side opposite to is alled a. a b Step 2. Using the method of Example 1, let us onstrut the triangle having the sides a, b, and so that a = a, b = b, =, goes along the given ray and the left end of oinides with the ray s vertex. a b b a 6

7 Sine the onstruted triangles are equal, the angle opposite to the side a must be equal to the angle opposite to the side a. a b b a 7

8 Solution 2. Let us draw a irumferene entered at the vertex of the original angle. The sides of the angle mark two points, A and B, on the line. Let us draw another irumferene of the same radius entered at the vertex of the given ray. Let us all A the point where the irumferene intersets the ray. Let us measure the distane between the points A and B with a ompass. Let us further stik the ompass s needle at A and mark the point B lying on the seond irumferene suh that AB = A B. B A B A The last thing to be done is to draw the ray originating at the enter of the seond irumferene and passing through B. 8

9 B A B A To prove that =, onsider the translation of the plane (a move of the plane parallel to itself) that shifts the enter of the seond irumferene to the enter of the first. This move will make the irumferenes oinide. Let us further rotate the seond angle until A oinides with A. Sine AB = A B, this move will make B oinide with B as well. Thus, the angles and are equal. 9

10 Problem 2 Using a ompass and a ruler, onstrut an angle equal to the angle below in two different ways. Use an auxiliary triangle on this page and the irumferenes on the next one. 10

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12 The word adjaent means lying near to as in adjaent rooms or the houses adjaent to the park. It was inherited from Latin without a hange in spelling. Problem 3 On the next page, draw a triangle with the angle and adjaent sides b and given below. In this ase, the word adjaent means that the vertex of is an endpoint of the sides b and. b Hint: begin with drawing the angle. If you hoose the auxiliary triangle method for onstruting the latter, you an use not some arbitrary sides b and, but the given ones right away! 12

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14 Problem 4 Draw a triangle with the side and adjaent angles and β given below. In this ase, the word adjaent means that the endpoints of are the verties of and β. β 14

15 Note that by arrying out the above onstrutions, we just have proven the following very important theorem. Theorem 1 Two triangles in the Eulidean plane are equal if either of the following holds. Their side lengths are pairwise equal. a = a, b = b, = They have an angle of equal size, and the lengths of the sides adjaent to the equal angles are pairwise equal. =, b = b, = They have a side of equal length, and the adjaent angles are pairwise equal. =, =, β = β Problem 5 On the next page, onstrut a triangle with the angle given below as well as the side b adjaent to and the side a opposite to the angle. b a 15

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