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# Brief outline of types of construction

By Roy Dunn,2014-04-18 01:15
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Brief outline of types of construction

Carpenter’s Geometric

Constructions

Kevin Mirus

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

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

and Q.

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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Bisecting angles

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

Q.

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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Step 7

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

the octagon.

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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.