Constructions
Math 366 Chapter 10 Constructions
Circle Construction
Given a center and a radius, 1) set the legs of the compass on the endpoints of the segment giving
the radius, 2) keeping the distance determined, set the compass pointer at the center and move
the pencil to draw the circle.
Segment Construction
Given a segment, construct a congruent segment by 1) setting the legs of the compass on the
endpoints of the given segment, 2) placing the point of the compass at one endpoint, and 3)
marking an arc to locate the other endpoint.
Constructing a Triangle Given Three Sides
Construct ∆ A’B’C’ ≅ ∆ ABC using the three sides.
B
A
C
1. Construct a segment A' C ' ≅ AC .
2. Construct a circle (or arc) with center at A’ and radius AB, and a circle (or arc) with
center at C’ and radius BC.
3. Label one of the intersection points of the two circles (or the intersection of the arcs) B’.
4. Draw A' B ' and B'C ' .
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Constructions
Constructing Congruent Angles
Construct ∠ A’B’C’ ≅ ∠ ABC
A
B
C
1.
2.
3.
4.
5.
Draw B ' C ' .
With center at B, mark off an arc AC .
Mark an arc with the same radius at center B’.
With pointer at C’, mark an arc C’A’, so that C’A’ = CA.
Draw B' A' .
Constructions Involving Two Sides and an Included Angle of a Triangle
Construct ∆ A’B’C’ ≅ ∆ ABC using two sides and the included angle.
B
A
C
1.
2.
3.
4.
Draw A' C ' , such that A' C ' ≅ AC .
Construct ∠ A’ ≅ ∠ A.
Mark B’ such that A' B ' ≅ AB .
Draw B'C ' .
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Constructions
Construction of the Perpendicular Bisector of a Segment
Construct a perpendicular bisector of AB .
A
1.
2.
3.
4.
B
Put the compass point on A and the pencil point anywhere past the midpoint.
Draw a circle (or arcs above and below) with A as center.
Draw the same size circle (or arcs above and below) with B as center.
Draw a segment, ray, or line through the two intersection points.
Construction of a Circle Circumscribed About a Triangle
A circle is circumscribed about a triangle when all three vertices of the triangle are on the circle.
The circle is called a circumcircle. Its center is called the circumcenter and its radius the
circumradius.
Construct a circle circumscribed about ∆ ABC.
B
C
A
1. Construct the perpendicular bisectors of AB and AC .
2. Construct a circle with radius DA , and center D, the intersection of the perpendicular
bisectors.
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Constructions
Constructing Parallel Lines
Rhombus method
Construct a line through P parallel to l.
P
l
1. Through P, draw any line that intersects l in A. PA will be a side of a rhombus.
2. Draw an arc with the pointer at A and radius AP to mark the third vertex, X, of a
rhombus.
3. With the same opening of the compass, draw intersecting arcs, first with the pointer at P
and then with the pointer at X to find Y, the fourth vertex of the rhombus.
4. Draw PY . PY // l in rhombus APYX.
Corresponding Angle method
Construct a line through P parallel to l.
P
l
1. Through P, draw a line that intersects l, forming angle α.
2. Copy angle α at point P. PY // l.
Constructing Angle Bisectors
Construct an angle bisector of ∠ A.
A
1. With the pointer at A, draw any arc
intersecting the angle at B and C, giving
three vertices of a rhombus with vertex at A.
2. Draw an arc with center at B and radius AB.
3. Draw an arc with center at C and radius AB. The arcs intersect at D, the fourth vertex of
the rhombus.
4. Connect A with D. AD is the angle bisector of A in rhombus ABDC. (The diagonals of a
rhombus bisect the angles.)
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Constructions
Constructing Perpendicular Lines
Construction of a Perpendicular to a Line through a Point not on the Line
Construct a line through P, perpendicular to line l (P is not a point on l).
P
l
1. Draw an arc with center at P that intersects the line at two points, A and B. (P, A, and B
will be vertices of a rhombus.)
2. With the same compass opening, make two intersecting arcs, one with center at A and the
other with center at B.
3. Connect P with Q, the intersection of the two arcs constructed in step 2. PQ is
perpendicular to l.
Construction of a Perpendicular to a Segment through the Midpoint of the Segment
This construction was done in section 10-1.
Construction of a Perpendicular to a Segment through a Point on the Segment
Construct a line through P, perpendicular to line l (P is a point on l).
P
l
1. Draw an arc with center at P that intersects l in two points, A and B. ( AB will be the
diagonal of a rhombus.)
2. Use a larger opening for the compass and draw intersecting arcs, with centers at A and B,
to determine C and D (endpoints of the other diagonal of the rhombus).
3. Connect C with D, the points where the arcs intersect. CD is perpendicular to l through
P. ( CD and AB are perpendicular bisectors of each other in rhombus ACBD.)
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Constructions
Constructing a Circle Inscribed in a Triangle.
Inscribe a circle in ∆ ABC.
B
A
C
1. Bisect the angles of the triangles. The intersection of the angle bisectors, P, will be the
center of the circle.
2. Construct a perpendicular from P to a side of the triangle. The length of that segment
will be the length of the radius of the circle.
Construction Separating a Segment into Congruent Parts
Separate AB into three congruent parts.
A
B
1. Draw any ray, AC , such that A, B, and C are noncollinear.
2. Mark off the given number of congruent segments (of any size) on AC . In this case, we
use three congruent segments.
3. Connect B to A3, the endpoint of the last of the three congruent segments.
4. Through A2 and A1, construct parallels to BA3 .
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