Geometric Construction

History of Euclid Euclidian Geometry (the kind we use in this class) was developed by a Greek mathematician name Euclid. Euclid lived from approx. 330 to 260bc and wrote a 13 volume book called Elements which illustrated all the concepts used in Geometric Construction.

“Why Didn‘t He Just Use a Ruler?” 5 ÷ 2 = ??? The Greeks could not do arithmetic!! They used Roman numerals, and so only had positive whole numbers (I, II, III, IV, V) - no negative numbers - no fractions or decimals -no zero So when measuring, if the length came out to anything other than an even number it could not be represented in Greek and Roman culture. As a result they had to use other tools such as a compass and straight edge. But they gave us some pretty good rules to follow to do it correctly.

REMEMBER: Where two lines intersect, a point is created!

Euclid’s 5 Laws for Constructions A straight line segment can be drawn joining any two points. Any straight line segment can be extended indefinitely in a straight line. Given a straight line segment, a circle can be drawn having the segment as radius and one endpoint as the center. All right angles are congruent to one another. If two lines are drawn that intersect a third line in such a way that the sum of the interior angles on the same side is less than two right angles, then the two lines, if extended, must intersect each other on that side. (“Parallel Postulate”) Mr. Witherspoon’s 3 Laws for Constructions: Draw constructions very lightly using guidelines. Do NOT erase your guidelines – this is how you show your work! Only trace over the final solution NOT the construction.

Don’t change your radius! Construction #1 Construct a segment congruent to a given segment. This is our compass. Given: A B Procedure: Draw two points A and B. Connect the points with a line segment. Construct: XY = AB 2. Draw point X. Use a straightedge to draw a line. (line l.) Don’t change your radius! 3. Set your compass for radius AB and make a mark on the line where B lies. Then, move your compass to line l and set your pointer on X. Make a mark on the line and label it Y. l X Y

Construct an angle congruent to a given angle Construction #2 Construct an angle congruent to a given angle A C B Given: Procedure: D 1) Draw a ray. Label it RY. 2) Using B as center and any radius, draw an arc intersecting BA and BC. Label the points of intersection D and E. E Construct: 3) Using R as center and the SAME RADIUS as in Step 2, draw an arc intersecting RY. Label point E2 the point where the arc intersects RY D2 R Y 4) Measure the arc from D to E. E2 5) Move the pointer to E2 and make an arc that that intersects the blue arc to get point D2 6) Draw a ray from R through D2