Basic Constructions and Points of Concurrency

Objectives What is construction? Who invented this tool commonly used in geometry? Circumcenter Incenter Centroid Orthocenter Euler Line

TOOLS NEEDED  COMPASS  STRAIGHT EDGE  PENCIL  PAPER  YOUR BRAIN (THE MOST IMPORTANT TOOL)

What do we mean by construction?  the drawing of geometric items such as lines and circles using only a compass and straightedge. Very importantly, you are not allowed to measure angles with a protractor, or measure lengths with a ruler.

Compass  The compass is a drawing instrument used for drawing circles and arcs. It has two legs, one with a point and the other with a pencil or lead. You can adjust the distance between the point and the pencil and that setting will remain until you change it.  Note: This kind of compass has nothing to do with the kind used find the North direction when you are lost. Straightedge  A straightedge is simply a guide for the pencil when drawing straight lines. In most cases you will use a ruler for this, since it is the most likely to be available, but you must not use the markings on the ruler during constructions. If possible, turn the ruler over so you cannot see them.

Father of Geometry  Euclid, the ancient Greek mathematician is the acknowledged inventor of geometry.  He did this over 2000 years ago, and his book "Elements" is still regarded as the ultimate geometry reference.

Why did Euclid do it this way?  Why didn't Euclid just measure things with a ruler and calculate lengths so as to bisect them?  The Greeks could not do arithmetic. They had only whole numbers, no zero, and no negative numbers like the Roman numerals. In short, they could perform very little useful arithmetic.  So, faced with the problem of finding the midpoint of a line, they could not do the obvious - measure it and divide by two. They had to have other ways, and this lead to the constructions using compass and straightedge. It is also the reason why the straightedge has no markings. It is definitely not a graduated ruler, but simply a pencil guide for making straight lines. Euclid and the Greeks solved problems graphically, by drawing shapes, as a substitute for using arithmetic.

POINTS OF CONCURRENCY

When two lines intersect at one point, we say that the lines are intersecting. The point at which they intersect is the point of intersection. Well, if three or more lines intersect at a common point, we say that the lines are concurrent lines. The point at which these lines intersect is called the point of concurrency. (nothing new right?) Concurrent Lines

Definitions Concurrent Lines – Three or more lines that intersect at a common point. Point of Concurrency – The point where concurrent lines intersect.

Perpendicular Bisector Both sides are congruent- make sure you see this or it is NOT a perpendicular bisector

Angle Bisector

Median Both sides are congruent

Altitude

Give the best name for AB AB ABAB ABAB || | | || Median Altitude None A Angle Perp Bisector Bisector

Perpendicular Bisectors The point where the 3 perpendicular bisectors meet is called the circumcentre. The circle which passes through the vertices of the triangle with the circumcentre as its centre, is called the circumcircle. A perpendicular bisector cuts a line exactly in half at a right angle.

Angle Bisectors xoxo xoxo An angle bisector is a line that cuts an angle exactly in half. yoyo yoyo zozo zozo xoxo xoxo The bisectors of the angles of a triangle are concurrent and the point of intersection is the centre of an inscribed circle.

Medians A median of a triangle is a line from a vertex to the mid-point of the opposite side. The point where the 3 medians meet is called the centroid.

Altitudes An altitude of a triangle is a line from a vertex perpendicular to the opposite side. The point where the 3 altitudes meet is called the orthocenter.

The orthocenter, circumcenter, and the centroid are COLLINEAR in EVERY triangle!

Points of Concurrency Concurrent Lines 3 or more lines that intersect at a common point Point of Concurrency The point of intersection when 3 or more lines intersect. Type of Line SegmentsPoint of Concurrency Perpendicular BisectorsCircumcenter Angle BisectorsIncenter MedianCentroid AltitudeOrthocenter

Points of Concurrency (con’t) Facts to remember: 1.The circumcenter of a triangle is equidistant from the vertices of the triangle. 2.Any point on the angle bisector is equidistant from the sides of the angle (Converse of #3) 3.Any point equidistant from the sides of an angle lies on the angle bisector. (Converse of #2) 4.The incenter of a triangle is equidistant from each side of the triangle. 5.The distance from a vertex of a triangle to the centroid is 2/3 of the median’s entire length. The length from the centroid to the midpoint is 1/3 of the length of the median.

Points of Concurrency (con’t)