Madison Area Technical College
Madison East High School Math Week
Monday, May 16, 2005
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2221. The Pythagorean Theorem: a + b = c (for right triangles)
Application: A triangle with sides of length 3, 4, and 5 (or multiples thereof) automatically forms a right triangle. This can be used to check that an angle is square,
or to lay out a square angle.
2. Copying line lengths
Problem: Construct a line segment congruent to a given segment.
Given: Line segment Step 1: Choose any Step 2: Use a compass
point C on line l. to measure the distance AB
AB, then draw an arc
centered at C with
radius AB that intersects
l at point D.
Comments: This may seem overly simple, but it has a lot of practical uses.
Application: Create an equilateral triangle, which has 60； angles.
Problem: Construct an equilateral triangle.
Application: You can use this to copy triangles, just by copying the lengths of each of the three sides. You can also use this to copy any polygon, since you can always divide it up into triangles.
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3. Constructing Perpendicular Lines
Application: To get the exact center of a distance without measuring, you can
construct its perpendicular bisector:
Problem: Construct the perpendicular bisector of a given line segment.
Step 1: Draw arcs centered Given: Line segment ABStep 2: Draw segment . PQ
on A and B that have the Label its intersection with same radius greater than the AB as point M. distance to the midpoint.
Label the intersections P
Comments: Constructing the perpendicular bisector gives you both the midpoint and a right angle!
Problem: Construct a line perpendicular to a given line and through a given point.
Starting with line segment AB and point C not on the line, strike an arc centered at C so that the arc intersects AB at two points (E and F). Construct the perpendicular bisector between EF, and it will pass through point C.
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Application: This technique allows you to make squares and rectangles.
Application: To find the center of a circle, pick any two random chords, then
construct the perpendicular bisectors of the chords. The bisectors will intersect at the
center of the circle.
Problem: Locate the center of a given circle.
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Application: Finding the center of an arch
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Problem: Construct the bisector of a given angle.
Given: Angle Step 1: Draw an arc of any radius centered ？BAC
at A. Label the points of intersection P and
Step 2: Draw arcs of equal radius from P and Q. The line from A through their
intersection bisects ？BAC.
Application: Making 45； and 22.5； angles from right angles, and making 30； and
15； angles from equilateral triangles.
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4. Other regular polygons
Application: Construct a regular pentagon.
Problem: Construct a regular pentagon.
Step 1 Step 2
Step 3 Step 4
Step 5 Step 6
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Problem: Construct a regular pentagon using a framing square.
The distance from B to P is the length of each side of the pentagon.
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Application: Construct a regular hexagon.
Problem: Construct a regular hexagon.
Just draw a circle, then mark the radius off six times around the circumference.
Comments: This forms the basis for many a church window. Extending the pattern makes a pretty tiling.
Application: Construct a regular Octagon… just make a square inside a circle, then
bisect the angles. Or, construct 22.5； angles off of line segments until you get around
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5. Parallel Line construction
Problem: Construct a line parallel to a given line through a point not on the given line.
Given line AB, and point C not on the line, draw line AC. Then, strike a random-length
arc centered at A that has intersection points D and E. Strike the same radius arc centered at C that has intersection point F. Use a compass to measure the distance from D to E, then mark off that same distance from point F to locate point G. The line through CG is parallel to AB.
Problem: Construct a line parallel to a given line through a point not on the
given line using the “draftsman's cheat”.
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