Regents Examination in Geometry: Constructions

BASIC CONSTRUCTIONS

Key Ideas
A construction uses a straightedge and a compass to create a precise geometric figure. The compass is used to construct arcs whose points are all equidistant from the center point. Intersecting arcs locate a point equidistant from the two centers. Two intersecting circles create points equidistant from the two centers. The fundamental loci described in Section 3.1 are used to create a variety of constructions from these two building blocks. Some of the basic constructions are the equilateral triangle, perpendicular line, and parallel line.


Definition of Construction
A construction is a geometric drawing in which only a straightedge and compass may be used. The straightedge is used only for connecting two points in a straight line. If your straightedge is a ruler, you cannot use the length markings on it to measure length. When constructing an angle of a particular measure, such as a 45° angle, you cannot use a protractor to measure the angle. The only measuring device allowed is the compass, and we will use it to measure both length and angle opening.

The Compass
The compass is simply a tool for constructing circles. The point of the compass locates the center point P, and the distance the compass is opened defines the radius. As long as the opening of the compass does not change, every point it traces out is a fixed distance from the center, resulting in a circle.
You should purchase a good-quality compass. A compass that is too loose or not rigid enough will be very frustrating to use.

Copy a Segment
Given:  and point C, construct  congruent to .
 

Procedure:
Place compass point on A and pencil on B; make a small arc.
With the same compass opening, place point on C and make an arc.
Use the straightedge to connect point C to any point D on the arc, .



Copy an Angle
Given: Angle ∠ABC and , construct ∠DEF congruent to ∠ABC.
 

Procedure:
Place compass point on B, and make an arc intersecting the angle at R and at S.
With the same compass opening, place point on E and make an arc intersecting  at T.
Place compass point on R and pencil on S, and make a small arc.
With the same compass opening, place point on T and make an arc intersecting the previous one at F.
Use the straightedge to form . ∠DEF is congruent to ∠ABC.



Equilateral Triangle
Given: Segment , construct an equilateral triangle with side length AB.
Procedure:
Place compass point on A and pencil on B. Make a quarter circle.
Place compass point on B and pencil on A. Make a quarter circle that intersects the first at C.
Use a straightedge to form AC and BC. ΔABC is an equilateral triangle.



Angle Bisector
Given: Angle ∠ABC, construct angle bisector .
Procedure:
Place compass point on B, and make an arc intersecting  and  at R and S, respectively.
Place compass point on R, and make an arc in the interior of the angle.
With the same compass opening, place compass point on S and make an arc that intersects the previous arc at point D.
Use the straightedge to form .
 is the angle bisector.



Perpendicular Bisector
Given: Segment , construct the perpendicular bisector of .
Procedure:
With the compass open more than half the length of , place the point at A and make a semicircle running above and below .
With the same compass opening, place the point at B and make a semicircle running above and below
. Make sure the semicircle intersects the first semicircle at R and at S.
Use a straightedge to connect R and S. 
is the perpendicular bisector of .

CONSTRUCTIONS THAT BUILD ON THE BASIC CONSTRUCTIONS

Key Ideas
The basic constructions of the previous section can be combined to produce a number of other constructions.

Perpendicular to Line from a Point not on the Line
Given line  and point P not on , construct a line perpendicular to  passing through P.
 

Procedure:
Place compass point at P, and make an arc intersecting  at R and S. (Extend  if necessary.)
Place compass point at R, and make an arc on the opposite side of the line as P.
With the same compass setting, place compass point at S, and make an arc intersecting the previous arc at Q.
Use the straightedge to connect P and Q.
 is perpendicular to  and passes through P.



Perpendicular to Line from a Point on the Line
Given line  and point P on  between A and B, construct a line perpendicular to  and passing through P.
 

Procedure:
Place compass point at P, and make an arc intersecting  at R and at S.
Place compass point at R, and make an arc below .
With the same compass setting, place point at S and make an arc intersecting the previous arc at Q.
Use a straightedge to connect P and Q.
 is perpendicular to  and passes through P.



Parallel to Line from Point off the Line
 

Given: Segment
 and point P not on , construct a line parallel to  and passing through P.
 

Procedure:
Use the straightedge to construct a line passing through P and intersecting  at R.
With compass point at R, make an arc intersecting  and  at S and T, respectively.
With the compass at the same setting and compass point at P, make an arc intersecting  at U.
With compass point at T, make an arc intersecting  at S.
With the compass at the same setting and compass point at U, make an arc intersecting at V.
Use a straightedge to connect P and V.
 is parallel to  and passes through point P.



Inscribe a Regular Hexagon or an Equilateral Triangle in a Circle
 

Procedure:
Construct circle P of radius PA.
Using the same radius as the circle, place compass point on A and make an arc intersecting the circle at B.
With the same compass setting, place compass point at B and make an arc intersecting the circle at C. Continue making arcs intersecting at D, E, and F.
Using the straightedge, connect each point on the circle to form regular hexagon ABCDEF. Connecting every other point will form equilateral triangle ACE.



Inscribe a Square in a Circle
 

Procedure:
Construct a diameter through the center point T of a circle.
Construct the perpendicular bisector of the first diameter, which will also be a diameter.
The intersections of the diameters with the circle are the vertices of the square.
Use a straightedge to connect the vertices and form the sides of the square.

POINTS OF CONCURRENCY, INSCRIBED FIGURES, AND CIRCUMSCRIBED FIGURES

Key Ideas
Angle bisectors, medians, altitudes, and perpendicular bisectors in triangles have the special property of being concurrent, which means they intersect at a single point.

Vocabulary
Some new vocabulary is needed to discuss points of currency in triangles:

Concurrent lines—3 or more lines that intersect at a single point.
Tangent—a line coplanar with a figure that intersects it at exactly one point, or a segment or ray that lies on a tangent line.
*(We also use tangent as an adjective, as in  is tangent to circle P.)
Figure shows  tangent to circle P.


 is tangent to circle P at Q.   is not a tangent.
 

Point of tangency—the point at which a tangent intersects a circle.
Inscribed circle (or incircle)—a circle that is tangent to each of the sides of a polygon.
Circumscribed circle (or circumcircle)—a circle that intersects each of the vertices of a polygon.

Incenter
The three angle bisectors of any triangle are concurrent at a point called the incenter. The incenter is the center of the circle tangent to each of the three sides of the triangle. This circle is called an inscribed circle. To construct the incenter of any triangle, simply construct the angle bisectors of at least two of the angles. Constructing the third would be a check since the first two will always intersect. If the third angle bisector intersects the first at the same point, then you know the construction is correct.

Figure shows the construction of the incenter of ΔABC.

Construction of the incenter of a triangle

Once the incenter is located, the point of tangency of the inscribed circle is located by constructing a line from the incenter perpendicular to one of the sides of the triangle. The theorem at work here is that a radius to a point on the circle is perpendicular to the tangent at that point. A circle can then be constructed using the incenter as the center and radius from the incenter to the point of tangency. Figure 3.3 shows the construction of the inscribed circle using the incenter.


Construction of the inscribed circle using the incenter

Since the incenter lies on the three angle bisectors, it is equidistant from the three sides of the triangle. The distance to each of the sides of the triangle is equal to the radius of the circle. The inscribed circle is also the largest circle that can fit inside a triangle, and the incenter is always inside the circle.

Math Fact
Every triangle can have an inscribed circle, but not every polygon with four or more sides can have one. The incenter of a triangle is equidistant from the three sides of the triangle.


Circumcenter
The three perpendicular bisectors of the three sides of any triangle are concurrent at a point called the circumcenter. The circumcenter is also the center of the circle that passes through each of the three vertices of the triangle. This circle is called a circumscribed circle. To construct the circumcenter, construct at least two of the perpendicular bisectors and find the point of concurrency. The circumscribed circle is centered at the circumcenter. The distance to any vertex of the triangle is the radius, as shown in Figure.


Construction of the circumcenter and circumscribed circle

The theorem at work here is that the points equidistant from two given points lie on the perpendicular bisector. The circumcenter is equidistant from each of the three vertices of triangle, so it must be the center of the circle containing those points.

Centroid
The three medians of a triangle are concurrent at a point called the centroid. Each median is constructed by first constructing the midpoint of each side of the triangle, using the perpendicular bisector construction. Then segments are drawn from each vertex to the midpoint of the opposite side.

The centroid is the intersection of the three medians. Figure shows the construction of centroid P and medians , , and  in ΔABC. The centroid of any triangle will always be located inside the triangle.

Construction of the median of a triangle


Math Fact
The centroid of a triangle is also the triangle’s center of gravity. The center of gravity is the point where the figure or object is perfectly balanced.

Orthocenter
The three altitudes of any triangle are concurrent at a point called the orthocenter. Figure below shows the construction orthocenter P of ΔABC. The orthocenter will be inside of an acute triangle, on a vertex of a right triangle, and outside an obtuse triangle.

Construction of the orthocenter, P, of triangle ΔABC

Points of Concurrency in Triangles

Segment	Where Concurrent	Feature
Perpendicular bisector	Circumcenter	Center of the circumscribed circle; equidistant from each of the three vertices
Angle bisector	Incenter	Center of the inscribed circle; equidistant from the three sides of the triangle
Median	Centroid	Divides each median in a 2 : 1 ratio; is the center of gravity
Altitude	Orthocenter	Located inside acute triangles, on a vertex of right triangles, and outside obtuse triangles

Math Tip
Mnemonic devices can be helpful for memorizing information. An example for points of concurrency is “all of my children are bringing in peanut butter cookies.”
AO—altitudes/orthocenter
MC—medians/centroid
ABI—angle bisectors/incenter
PBC—perpendicular bisectors/circumcenter