Similar fioures

See on the Fig 29. The two figures look alike but one is a smaller copy of the other. These are similar figures: figures having the same shape but not necessary the same size.

The sides in the same position are the corresponding sides of the figures, and the angles in the same position are the corresponding angles of the figures. For example, A'B'(read A prime B prime) are corresponding sides in the figures above, and ABD and A'B'D' are corresponding angles. The corresponding angles of similar polygons are equal.

If the ratio of side AB to side A'B' is 2:1, then the ratio of a pair of any other two corresponding sides must also be 2:1. The corresponding sides of similar polygons will be parallel if the polygons are placed in the same position.

Rule to Remember: Two polygons are similar if two conditions are satisfied:

1) corresponding sides are proportional;

2) responding angles are equal.

Exercises

I. Read the following words paying attention to the pronunciation:

dimension, condition, slide, alike, line, similar, equal, , that, those, this, they.

2. Make up sentences of your own using the words and expressions given below:

to look alike, in the same position, but not necessarily, corresponding sides.

III. Answer the following questions:

1. What figures do we call similar ones? 2. What angles of similar polygons are equal? 3. When are two polygons similar?

IV. Translate into Russian:

Whenever you draw a map or a plan to scale, you draw a figure similar to the original in shape but different in size. When we reduce the size of a photograph, the smaller picture is similar to the larger one.

In similar triangles the ratio of any pair of corresponding sides equals the ratio of either of the other pairs; that is, they are proportional. In similar triangles the corresponding angles are equal.

Trigonometry

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Trigonometry and its application

Trigonometry is a branch of mathematics that deals with angles and sides of triangles and their relations to one another. Otherwise we can say that trigonometry is the set of methods and procedures required to solve problems concerning trian­gles when angles of the triangles are involved. It is especially useful in numerous calculations -connected with accurate machine work. It is also very useful to the surveyor, draughts­man and, in fact, is used in all sorts of engineering work. The carpenter with his steel square makes use of trigonomet­ric relations to find the length of one side of hip roof.

Many problems about alternating current can be solved by using trigonometry. In meteorology, the height of a balloon above the surface of the earth is determined by the use of trigonometry.

Trigonometry has applications in surveying, navigation, construction work and many branches of science. It is particularly essential for most branches of mathematics and physics.

Exercises

I. Read the following words paying attention to the pronuncia­tion:

calculation, relation, ratio, angle, triangle, root, roof, balloon, use, cube, deal, meeting.

II. Make up sentences of your own using the words and expres­sions given below:

to be useful to, in engineering work, otherwise, branch of mathematics, branch of physics, to deal with angles, to deal with; triangles, to deal with relations.

III. Answer the following questions:

1. What does trigonometry deal with? 2. In what work is trigonometry especially useful? 3. For what purpose does the carpenter make use of trigonometric relations? 4. What is determined in meteorology by the use of trigonometry?

IV. Translate into Russian:

A branch of mathematics dealing with the relationship between the sides and angles of triangles is called trigonom­etry. It is defined as the branch of mathematics using the fact that numerous problems may be solved by the calculation of unknown parts (sides and angles). The solution of such problems is greatly assisted by the use of the trigonometrical ratios or functions.

V. Translate into English:

С помощью тригонометрии решаются многие измери­тельные задачи на местности, как например, вычисление расстояний между различными пунктами земной поверх­ности (earth surface), вычисление высоты данного предмета, составление карт и т.п. Измерение небольших расстояний производится при помощи стальных измерительных лент (steel measuring tapes). Измерение углов производится при помощи угломерных инструментов (goniometrial instru­ments).

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