1 Basic concepts in geometry

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1 1 asic concepts in geometry 1.1 Introduction We start geometry with the simpest idea a point. It is shown using a dot, which is abeed with a capita etter. The exampe above is the point. straight ine is a set of points next to each other. In maths, it carries on in both directions to infinity even if we don t draw it, we think of it going on for ever in two opposite directions. It is shown either using two points (abeed with capita etters) on a ine with a doube headed arrow, or using a singe ower case etter. The ine, and the ine. pane is a fat surface extending in a directions to infinity. It has no thickness, so it is two dimensiona. pane has no size or shape, but it can be shown using a quadriatera abeed with a singe capita etter. Some of the ideas about points, ines and panes can be given as axioms (statements given without proof). 1. n infinite number of ines can be drawn through any given point. 2. One and ony one ine can be drawn through two distinct points. 3. When two ines intersect they do so at ony one point. 1-1

2 1.2 Coinear and Copanar ny two points wi aways ie on the ine drawn between them, but three points do not have to if they do, they are caed coinear points. It the three or more points are not on the same ine, they are non coinear. C R m Line passes through points, and C, so, and C are coinear. It is impossibe to draw a straight ine through a three of, and R, so they are noncoinear or just non inear. Simiary, points and ines which ie in the same pane are caed copanar, otherwise they are caed non-copanar. There are severa specia cases of ines. ine segment is the straight path between two points (e.g. C and D). It is named using its two endpoints. It is written CD and can be read as segment CD. s said above, a ine has no end points. It extends forever in both directions. If the points M and N are on the ine, it can be written as MN and read as ine MN. ray is part of a ine with one end point. It is infinite in the other direction. It is written as RS which is read as ray RS where R is the end, and S is another point on the ine. ecause a pane as defined above is infinite in a directions, it can contain any number of ines, rays, ine segments and points. (What geometric idea does a torch suggest?) 1-2

3 1.3 Here are some axioms about panes: xiom: pane can be defined using a ine and one point not on the ine. y using the definition of a ine, a pane can contain three non-coinear points. Conversey, through any three non coinear points there can be one and ony one pane. C So ine and point C are contained in the same pane., and C are three non-coinear points through which one pane, the pane can pass. xiom: If two ines intersect, exacty one pane passes through both of the ines. m ane contains intersecting ines and m. xiom: If two panes intersect their intersection is exacty one ine. anes and intersect and their intersection is ine. xiom: If a ine does not ie in a pane, but intersects it, their intersection is a point. oint is the intersection point of ine and pane. 1-3

4 Exampe 1 Draw three non-coinear points, and C on paper. How many different ines can be drawn through pairs of points? Name the ines. Soution Three ines can be drawn,, C and C C Exampe 2 D C S R For the above figure, answer the foowing (assuming a anges are 90 ): a. Name the ines parae to ine. b. re ine and point R copanar? Why? c. re points, S, and R copanar? Why? d. Name three panes passing through. Soution a. CD, SR and. b. Yes. ny ine and a point outside it are copanar. c. Yes. Two parae ines are aways copanar. d. CD, DS and DC. 1-4

5 1.4 Line Segments ine segment is a part of a ine. It has a fixed ength and consequenty two end points. They are used to name the ine segment The ine stretches to infinity in both directions, but the ine segment is finite, and starts at and stops at. ine contains an infinite number of ine segments. If two ine segments have a common end point, they can be added. M R For the above ine, M and MR are two segments. They have a common end point, M. Therefore, M + MR= R The midpoint of a ine segment is the point which is an equa distance from the two endpoints. If M is the midpoint of the ine segment R, then M + MR= R and M = MR, so R= 2 M = 2MR. Every segment has one and ony one midpoint. Exercise re the foowing statements true or fase? 1. ny number of ines can pass through a singe given point. 2. If two points ie in a pane the ine joining them aso ies in the same pane. 3. ny number of ines can pass through two given points. 4. Two ines can intersect in more than one point. 5. Two panes intersect to give two ines. 6. If two ines intersect ony one pane contains both the ines. 7. ine segment has two end points and hence a fixed ength. 8. The distance of the midpoint of a ine segment from one end may or may not be equa to its distance from the other. 1-5

6 Vocabuary point a pace in space, which has no size, it is infinitey sma ine a set of points stretching to infinity in both directions, but with no width. pane a mathematica fat surface in two dimensions, i.e. with no thickness axiom a ogica/mathematica statement which is given without proof as it shoud be obvious intersect cross (ike an intersection is merican Engish for where roads cross) coinear three or more points on the same ine non-coinear three or more points which are not on the same ine ray a ine with one fixed end, infinite in the other direction. conversey an argument or statement made the other way round from the origina. 1-6