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Machine Tool Math 2
Lab #1: Geometric Constructions Not In Your Book
Assigned:
Due:
Tuesday 1/27/2009
Tuesday 2/03/2009
Score = ___ / 165
1. Perpendicular Through a Point NOT on a Line
( __ / 10 ) Use a compass and straight edge only to construct the perpendicular of line m through point
C below.
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2. Parallel Through a Point NOT on a Line
( __ / 10 ) Use a compass and straight edge only to construct the parallel of line m through point C
below.
3. Tangent Line to a Circle from a Given Point on the Circle
( __ / 10 ) Use a compass and straight edge only to construct the tangent to circle O at point X on the
diagram below.
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4. Finding the Center of a Circle
( __ / 10 ) Suppose you are given a circle and you need to find its center. You can do this by drawing
any two random chords on the circle, like AB and CD in the diagram below. Then you construct the
perpendicular bisectors of each segment. The perpendicular bisectors will intersect at the center of the circle,
since the perpendicular bisectors of a chord pass through the center of a circle. Use a compass and straight
edge only to find the center of the circle on the right. Check your work by seeing if a circle drawn from that
center matched up with the original circle.
5. Circle Externally Tangent to Another Circle
( __ / 10 ) Use a compass and ruler only to construct a circle of radius 3 cm that is externally tangent to
circle O at point X on the diagram below. Hint: The center of the tangent circle will lie on line OX and be 3 cm
away from the edge of circle O.
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6. Circle Internally Tangent to Another Circle
( __ / 10 ) Use a compass and ruler only to construct a circle of radius 2 cm that is internally tangent to
circle O at point X on the diagram below.
7. Circle Externally Tangent to Two Other Circles
( __ / 10 ) Use a compass and ruler only to construct a circle of radius 3 cm that is externally tangent to
both circles in the diagram below.
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8. Circle Internally Tangent to An Angle
( __ / 10 ) Use a compass and ruler only to construct a circle of radius 2 cm that is internally tangent to
the angle in the diagram below. Hint: The center of the circle will be 2 cm away from each side of the angle.
9. Special Segments of Triangles: Medians
( __ / 10 ) Use a compass and straight edge only to construct all three medians of ABC below. A
median connects a vertex of a triangle with the midpoint of the opposite side. What appears to happen with
the medians?
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10. Special Segments of Triangles: Altitudes
( __ / 10 ) Use a compass and straight edge only to construct all three altitudes of ABC below. An
altitude is a line through a vertex that is perpendicular to the opposite side. What appears to happen to the
altitudes?
11. Special Segments of Triangles: Angle Bisectors and the Incircle
( __ / 10 ) Use a compass and straight edge only to construct all three angle bisectors of ABC below.
Draw a circle (called the incircle) that is centered where the three angle bisectors intersect and that has a
radius that is tangent to each of the three sides of the triangle.
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12. Special Segments of Triangles: Perpendicular Bisectors and the Outcircle
( __ / 10 ) Use a compass and straight edge only to construct all three perpendicular bisectors of
ABC below. Draw a circle (called the outcircle) that is centered where the three perpendicular bisectors
intersect and that has a radius that just touches each of the three vertices of the triangle.
Regular Polygons From a Rosette
A beautiful and useful geometric shape called a rosette can be drawn by first drawing a single circle, and
then keeping the compass set to the radius of that circle, mark off six equal segments around the original
circle. If you draw entire circles instead of just marking the arcs, you get the rosette pattern:
13. ( __ / 5 ) Use a six point rosette to construct an equilateral triangle.
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14. ( __ / 5 ) Use a six point rosette to construct a regular hexagon.
15. ( __ / 5 ) Use a six point rosette to construct a dodecagon (twelve-sided polygon).
16. ( __ / 5 ) Use the dodecagon to construct a square.
17. ( __ / 5 ) Use the square (in a rosette) to construct an octagon.
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A Regular Pentagon Construction: A compass and straight edge can be used to construct a pentagon
according to the instructions below.
a. Draw two perpendicular lines.
b. Draw a circle centered where the lines intersect. Label the point where the circle crosses the
vertical line as B and where it crosses the horizontal line as C.
c. Keeping the same radius on the compass, place its tip at point C, and strike an arc that intersects
the circle twice.
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d. Draw a line connecting those two intersection points, and label the point where it crosses the
horizontal line as D.
e. Place the tip of the compass at D, and draw an arc that starts at B and ends when the arc crosses
the horizontal axis. Label that new intersection point E.
f. Place the tip of the compass at B, and draw an arc that passes through E and intersects the circle
twice.
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g. Draw line segments that connect B to the two new intersection points on both sides of B.
h. Use the compass to construct congruent line segments around the circle to the segments drawn in
the previous step to complete the pentagon.
18. ( __ / 10 ) Use the space below to construct your own regular pentagon.
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19. Six Circles Internally Tangent to a Given Circle
( __ / 10 ) Use a compass and straight edge only to construct six circles externally tangent to a common
inner circle and internally tangent to a common outer circle. Use the large circle below for the common
outer circle. Your final construction should look like the small diagram. This pattern is common in stained
glass windows in many cathedrals because it symbolizes the 6 common days of the week “revolving” about
the Sabbath (inner circle).