“Geometric Constructions”- From the Foundation of Euclidean Geometry
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“Geometric Constructions”- From the Foundation of Euclidean Geometry
“Geometric Constructions”
From the Foundation of Euclidean Geometry
By
Don Beatty
TEC4619
Module Lesson Plan
June 12, 2009
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“Geometric Constructions”- From the Foundation of Euclidean Geometry
The North Carolina Department of Public Instruction (NCDPI)has identified the following criteria to be used for teaching Geometric Constructions in Drafting I. The following outline was taken from this active link to the DPI website.
DBasic geometric terms and constructions
004 Explain geometric terms and apply geometric construction techniques.
004.01Explain selected geometric terms.
004.02Explain the procedures for drawing standard geometric constructions.
004.03Construct drawings that require geometric constructions.
For the next two weeks we will develop a greater understanding of the usefulness and hands-on development of geometric constructions.
Introduction to Euclid and his Geometric Constructions
Euclid was a Greek mathematician, born around 300 BC and widely considered the “Father of Geometry”. He wrote a book on number theory and mathematical proofs, titledElements, which may be the most influential and useful textbook ever written. Elementsis a collection of axioms, theorems and proofs about squares, circles acute angles, isosceles triangles, and similar geometric identities. Most of the theorems which are taught in high schools today can be found in Euclid's 2000 year old book.
"Euclid's Elements - All thirteen books in one volume" is still available and is in print by Green Lion Press. ISBN 1-888009-18-7.
In this module of instruction, we will be discussing the application and development of geometric constructions. Although geometric constructions were around before Euclid, he was the first to organize them and apply theorems to validate their accuracy. Many geometric constructions are part of the more encompassing “Euclidean Geometry”.
Tools
Compass-A technical drawing instrument used for transferring measurements and drawing circles and arcs. (See photo right.)
Straight edge – Any tool with an accurately straight edge used for drawing straight lines. (Although a ruler may be used, its divisional markings may not be.)
Geometric Constructionsare created using only a compass and straight edge. With these instruments we will be able to:
Section 1
1Bisect a line
2Bisect an angle
3Draw a perpendicular line at the endpoint of a given line
4Draw a line parallel to a given line (non-Euclidean)
5Replicate an angle or triangle
6Construct 30°, 45° and 60° angles
7Draw an equilateral triangle of given side length.
Section 2
1Draw a circle through three given points
2Find the center of a circle
3Draw a tangent line at a given point on a circle
Section 3 (Non-Euclidean Geometry)
1Divide a line into equal parts
2From a given circle:
- Inscribe an equilateral triangle
- Inscribe a square
- Inscribe an equilateral pentagon
- Locate the center of a circle using any right-angle object
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“Geometric Constructions”- From the Foundation of Euclidean Geometry
Terms (NCDPI Competency 004.01)
- Point – Zero dimensional object representing a location in space
- Circle – A 2 dimensional figure consisting of a set of points in a plane equidistant from a given (center) point
- Arc – A portion of a circle
- Line – Set of points, defined by two given points, which has no ends
- Ray – Figure defined by one point and a direction from that point
- Angle–Figure created by two rays, lines, line segments or planes that share a common point
- Triangle – Polygon with three sides and three angles
- Square – Polygon with four equal length sides and four 90 degree angles
- Pentagon – Polygon with 5 sides and five angles
- Bisect – Cut or divide into two parts. (In our case, equally)
- Transfer – Draw an exact copy of a geometric object. (Think copy and paste.)
- Tangent – Condition where a straight line touches a curve at only one point without crossing there.
- Inscribe – to draw a geometric figure completely within another touching at extreme points only. In our case we may inscribe a polygon in a circle, or inscribe a circle in a polygon.
- Circumscribe - to draw the smallest possible geometric figure completely around another, touching at extreme points only. In our case we may circumscribe a polygon about a circle, with lines tangent, or circumscribe a circle about a polygon touching only at the points.
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“Geometric Constructions”- From the Foundation of Euclidean Geometry
Euclid’s Postulates
Euclid divided his ten axioms, which he called "postulates," into two groups of five. The first five were "Common Notions," because they were common to all sciences:
- Things which are equal to the same thing are also equal to one another.
- If equals are added to equals, the sums are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things which coincide with one another are equal to one another.
- The whole is greater than the part.
The remaining five postulates relate specifically to geometry:
- Any two points can be joined by a straight line.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All right angles are congruent.
- If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.
Euclidean geometry is distinguished from other geometries by the parallel postulate, which is more easily phrased as follows “Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.”
Procedures for Drawing Geometric Constructions (NCDPI Competency 004.02)
Section 1
1Create a Perpendicular Bisector of a line (segment)**
Step 1 / Place the compass on one end of the line segment.Step 2 / Set the compass width to approximately two thirds the length of the line. The actual width does not matter.
Step 3 / Without changing the compass width, draw an arc above and below the line.
Step 4 / Again without changing the compass width, place the compass point on the other end of the line. Draw an arc above and below the line so that the arcs cross the first two.
Step 5 / Using a straightedge, draw a line between the points where the arcs intersect.
Step 6 / This line is perpendicular to the first line and bisects it (cuts it at the exact midpoint of the line).
2Bisect an angle **
Step 1 / Place the compass point on the angle's vertex “Q”.Step 2 / Adjust the compass to a medium wide setting. The exact width is not important.
Step 3 / Without changing the compass width, draw an arc across each leg of the angle.
Step 4 / The compass width can be changed here if desired. Recommended: leave it the same.
Step 5 / Place the compass on the point where one arc crosses a leg and draw an arc in the interior of the angle.
Step 6 / Without changing the compass setting, repeat for the other leg so that the two arcs cross.
Step 7 / Using a straightedge, draw a line from the vertex to the point where the arcs cross
Step 8 / Done. This is the bisector of the angle PQR.
3Draw a perpendicular line at the endpoint of a given line or ray **
Step 1 / Pick a point not on the line, about 6 cm from the endpoint of the line or ray. Its exact location is not important. Label it D.Step 2 / Set the compass on point D and set its width to the endpoint of the line or ray.
Step 3 / Draw an arc that crosses the line or ray and extends over and above the ray endpoint. (If you prefer, draw a complete circle.)
Step 4 / Draw a diameter through D from the point where the arc crosses the line or ray.
Step 5 / Draw a line from the line or ray's endpoint to the endpoint of the diameter line
Step 6 / Done. The last line drawn is perpendicular to the line or ray.
4A. Draw a line parallelto a given line
Step 1 / Follow instructions for construction 3 above to create an additional perpendicular line at any other point on the line or ray (farther away from original is better) Label it E.Step 2 / Set the compass on point D and set its width to the desired distance between lines
Step 3 / Draw an arc that crosses the perpendicular line
Step 4 / Set the compass on point E and set its width to the desired distance between lines
Step 5 / Draw an arc that crosses the perpendicular line
Step 6 / The points defined by the intersecting arcs and perpendicular lines will define the new, parallel line.
-OR-
B.Draw a parallel line through a given point **
Step 1 / Draw a transverse line through given point R and across the line PQ at an angle, forming the point J where it intersects the line PQ. The exact angle is not important.Step 2 / With the compass width set to about half the distance between R and J, place the point on J, and draw an arc across both lines.
Step 3 / Without adjusting the compass width, move the compass to R and draw a similar arc to the one in step 2.
Step 4 / Set compass width to the distance between where the lower arc crosses the two lines
Step 5 / Move the compass to where the upper arc crosses the transverse line and draw an arc across the upper arc, forming point S.
Step 6 / Draw a straight line through points R and S.
Step 6 / Done. The line RS is parallel to the line PQ
5Transfer (Reproduce) an angle or triangle **
Step 1 / Mark a point P that will be one vertex of the new triangle (or angle)Step 2 / Set the compass width to the length of one side of the original (tri) angle ABC.
Step 3 / With the compass point on P, make an arc near the next vertex of the (tri) angle.
Step 4 / Mark a point R on the arc. This will become the next vertex of the new (tri) angle. PR is equal in length to AC
Step 5 / Use the compass to measure the length of the side AB in the original (tri) angle.
Step 6 / Place the compass point on P and make an arc in the vicinity of where the third vertex of the triangle will be. All points along this arc are the distance AB from P, but we do not yet quite know exactly where the vertex is.
Step 7 / Use the compass to measure the length of the side BC in the original (tri) angle
Step 8 / From point R, draw an arc crossing the first. Where these intersect is the vertex Q of the (tri) angle. (Stop here if only creating an angle. Continue to finish a triangle)
Step 9 / Finally, draw the three sides of the new triangle PQ ,PR, and QR.
Step 10 / Done. The new triangle PQR is congruent to the triangle ABC.
6Construct 30°, 45° and 60° angles
30° angle **
Step 1 / Draw a line segment which will become one side of the angle. (Skip this step if you are given this line.) The exact length is not important. Label it PQ. P will be the angle's vertex.Step 2 / Set the compass on P, and set its width to any convenient setting.
Step 3 / Draw an arc across PQ and up over above the point P. Label the point where it crosses PQ as point S.
Step 4 / Without changing the compass width, move the compass to the point S. Draw a broad arc that crosses the first one and goes well to the right. Label the point where the two arcs cross as point T.
Step 5 / Without changing the compass width, move the compass to the point T, and draw an arc across the previous arc, creating point R.
Step 6 / Draw a line from P to R.
Step 7 / Done. The angle QPR has a measure of 30°
45° angle **
Step 1 / Draw a line segment which will become one side of the angle. (Skip this step if you are given this line.) The exact length is not important. Label it PQ. P will be the angle's vertex.Step 2 / Set the compass width to just over half the length of the line segment PQ.
Step 3 / With the compass point on P then Q, draw two arcs that cross above and below the line.
Step 4 / Draw a line between the two arc intersections. This is at right angles to PQ and bisects it (divides it in exactly half).
Step 5 / With the compass point on the intersection of PQ and the perpendicular just drawn, set the compass width to P
Step 6 / Draw an arc across the perpendicular, creating the point C
Step 7 / Draw a line from P through C, and on a little more. The end of this line is point R
Step 8 / Done. The angle ∠QPR has a measure of 45°
60° angle **
Step 1 / Draw a line segment which will become one side of the angle. (Skip this step if you are given this line.) The exact length is not important. Label it PQ. P will be the angle's vertex.Step 2 / Set the compass on P, and set its width to any convenient setting.
Step 3 / Draw an arc across PQ and up over above the point P.
Step 4 / Without changing the compass width, move the compass to the point where the arc crosses PQ, and make an arc that crosses the first one.
Step 5 / Draw a line from P, through the intersection of the two arcs.
Step 6 / Done. The angle QPR has a measure of 60°
7Draw an equilateral triangle of given side length. **
Step 1 / Pick a point P that will be one vertex of the finished triangle.Step 2 / Place the point of the compass on the point A and set it's drawing end to point B. The compass is now set to the length of the sides of the finished triangle. Do not change it from now on.
Step 3 / With the compass point on P, make two arcs, each roughly where the other two vertices of the triangle will be.
Step 4 / On one of the arcs, mark a point Q that will be a second vertex of the triangle. It does not matter which arc you pick, or where on the arc you draw the point.
Step 5 / Place the compass point on Q and draw an arc that crosses the other arc, creating point R.
Step 6 / Using the straightedge, draw three lines linking the points P,Q and R.
Step 7 / Done. The triangle PQR is an equilateral triangle. Its side length is equal to the distance AB.
Section 2
1Draw a circle through three given points **
Step 1 / Draw straight lines to create the line segments AB and BC. Any two pairs of the points will work.Step 2 / Find the perpendicular bisector of one of the lines. (See Constructing the Perpendicular Bisector of a Line Segment.)
Step 3 / Repeat for the other line.
Step 4 / The point where these two perpendicular lines intersect is the center of the circle we desire.
Step 5 / Place the compass point on the intersection of the perpendiculars and set the compass width to one of the points A,B or C. Draw a circle that will pass through all three.
Step 6 / Done. The circle drawn is the only circle that will pass through all three points.
2Locate the center of a given circle
Step 1 / Using a straightedge, draw any two chords of the circle. For greatest accuracy, avoid chords that are nearly parallel. (They may cross)Step 2 / Construct the perpendicular bisector of one of the chords using the method described in Constructing a perpendicular bisector of a line segment (section 1 construction 1)
Step 3 / Repeat for the other chord
Step 4 / The point where the two lines intersect is the center C of the circle.
3Draw a tangent line to a given circle at a point
Step 1 / Draw a straight line through the center O of the circle and the point P right across the circle. This is a diameter of the circle.Step 2 / Mark a point Q anywhere. For best accuracy, avoid putting it too close to the diameter line.
Step 3 / Place the compass on the point Q just drawn, and set it's width to the point P.
Step 4 / Without changing the width, draw an arc across the diameter line, creating point R.
Step 5 / Without changing its width, draw another arc on the opposite side of Q.
Step 6 / Using the straightedge, draw a line through R and Q, extending it onwards so it crosses the arc just drawn. Mark this point S.
Step 7 / Using the straightedge, draw a line through P and S, extending it in both directions.
Step 8 / Done. The line just drawn is the tangent to the circle O through point P.
Section 3(Non-Euclidean Geometry)
1Divide a line into equal parts **
These instructions divide the line into “5” parts. Substitute the desired number in the appropriate places.