Download Geometry Collateral Constructions

Townsend Harris HS
Geometry
Name ___________________
Date______________
Geometry Collateral Constructions
1. Find the circumcenter of a triangle. Follow these steps:
a. Demonstrate once how to construct the  bisector of a line segment.
b. Draw a large scalene triangle.
c. Construct the  bisectors of two sides of the triangle. Draw the bisector lines.
d. Label their point of intersection as point P.
e. P is the center of the circumscribed circle.
f. With the compass, measure the distance from P to any vertex of the triangle.
This distance is the radius of the circle.
g. Draw the circumscribed circle with the compass.
h. Repeat steps b to g one more time.
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2. Find the incenter of a triangle. Follow these steps:
a. Demonstrate once how to bisect an angle.
b. Demonstrate once how to construct a line  to a given line through a point not
on the line.
c. Draw a large scalene triangle.
d. Bisect any 2 angles of the triangle. Draw the bisectors and label their point of
intersection P. (This is the center of the circle and the incenter.)
e. Construct a line  to one side of the triangle, through point P. Label the point,
where the  line touches the side of the triangle, A.
f. Measure the distance between points A and P, using the compass. (This is the
radius of the circle.)
g. Draw the inscribed circle using P as the center and AP as the radius.
h. Repeat steps c to g one more time.
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3. Find the centroid of a triangle. Follow these steps:
a. Demonstrate once how to construct the  bisector of a line segment.
b. Draw a large scalene triangle.
c. Construct the  bisectors of two sides of the triangle. Draw the bisector lines.
d. Label the points where the bisector lines cross the sides of the triangle as A
and B.
e. Draw the median from the opposite vertex to point A. Do the same for point
B. Label the point of intersection of the medians point P. This point is the
centroid of the triangle.
f. Repeat steps b to e one more time.
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4. Find the orthocenter of a triangle. Follow these steps.
a. Demonstrate once how to construct a line  to a given line through a point not
on the line.
b. Draw a large scalene triangle.
c. Construct a line  to one side of the triangle, through the vertex opposite the
side. Draw the  line.
d. Repeat step c, using a different side of the triangle.
e. The point where these two  lines meet is the orthocenter of the triangle.
f. Repeat steps b to e one more time.
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5. Construct a parallelogram. Follow these steps:
a. Demonstrate once how to construct a line parallel to a given line through a
given outside point using congruent corresponding angles.
b. Draw a large acute angle.
c. Call the angle ABC, where B is the vertex.
d. Construct a line parallel to side BC, using point A. (Extend side AB through
point A, to construct congruent corresponding angles at vertex A and vertex
B.) Draw the parallel line.
e. Measure side BC with a compass. Mark off the length of BC on the parallel
line constructed in part d.
f. Complete the parallelogram.
g. Repeat steps b to f one more time.
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6. Construct a 30 angle. Follow these steps:
a. Demonstrate once how to bisect an angle.
b. Construct a large circle. Label the center and any one point on the circle.
c. With the compass, measure the distance from the point to the center of the
circle. This is the radius of the circle.
d. Begin at the point on the circle. Mark off the (radius) distance 6 times around
the circle. Explain that if these points were connected, the resulting figure
would be a hexagon. (Do not connect these points.)
e. With a ruler, connect every other point on the circle, to form an equilateral
triangle.
f. Bisect one angle of the triangle to get the 30 angle.
g. Repeat steps b to f one more time.
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7. Construct a 45 angle. Follow these steps:
a. Demonstrate once how to construct a line  to a given line through a point on
the line.
b. Demonstrate once how to bisect an angle.
c. Draw a line and place a point on the line.
d. Construct a line  to the line drawn in part c, through the point on the line.
e. Draw the perpendicular line.
f. Construct the bisector of one of the right angles formed to get a 45 angle.
g. Repeat steps c to f one more time.
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8. Construct a triangle similar to a given triangle on a given line segment as a base.
a. Demonstrate once how to copy an angle.
b. Draw a large scalene triangle and call it ΔABC.
c. Draw a line segment RS (smaller in length than side AB of the triangle).
d. Line segment RS corresponds to side AB of ΔABC.
e. At point R, construct <SRL congruent to <BAC.
f. At point S, construct <RSM congruent to <ABC.
g. Label the point of intersection of rays RL and SM, as point T.
h. ΔRST is similar to ΔABC.
i. Repeat steps b to h one more time.
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9. Construct a line tangent to a given circle through a given point on the circle.
a. Demonstrate once how to construct a line perpendicular to a given line
through a point on the line.
b. Draw a circle and label the center point as O.
c. Label a point on the circle as point P.
d. Draw radius OP and extend it outside the circle.
e. Construct a line  to radius OP through point P. Call it line RP.
f. RP is a line tangent to circle O since the tangent and the radius of the circle
are perpendicular to each other.
g. Repeat steps b to f one more time.
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