11 January 2019Mr. Feist’s Geometric Construction CookbookPage 1 of 61
Mr. Feist’s
Mr. Bruce Feist
December, 2004
Revised 5/1/2012 11:42 PM
This guide demonstrates how to do various geometric constructions. Often a construction refers back to directions on how to do an earlier one. The guide includes practice worksheets for each construction, and can be used both as a reference and as a workbook.
11 January 2019Mr. Feist’s Geometric Construction CookbookPage 1 of 61
Contents
Segment Congruent to a Segment......
Segment Congruent to a Segment Practice Worksheet......
Angle Congruent to an Angle......
Angle Congruent to an Angle Practice Worksheet......
Perpendicular Bisector of a Segment......
Perpendicular Bisector of a Segment Practice Worksheet......
Perpendicular through a Point......
Perpendicular Through a Point Practice Worksheet......
Parallel Through a Point......
Parallel Through a Point Worksheet......
Angle Bisector......
Angle Bisector Practice Worksheet......
Circumscribe a Circle around a Triangle......
Circumscribe a Circle around a Triangle Practice Worksheet......
Inscribe a Circle within a Triangle......
Inscribe a Circle within a Triangle Practice Worksheet......
Median of a Triangle......
Median of a Triangle Practice Worksheet......
Centroid of a Triangle......
Centroid of a Triangle Practice Worksheet......
Orthocenter of a Triangle......
Orthocenter of a Triangle Practice Worksheet......
Midsegment of a Triangle......
Midsegment Practice Worksheet......
Reflect a Point through a Line......
Reflection Practice......
Rotate a Point Around a Point By an Angle
Rotation Practice
Glossary......
Segment Congruent to a Segment
Given: Segment AB
Goal: Construct a segment on another line which is congruent to AB.
Plan:
1)Measure segment AB with the compass.
2)Draw a circle centered on C with radius AB. Call the intersections D and E.
Segments CD and CE are both congruent to segment AB.
1)Measure segment AB with the compass.
2)Draw a circle centered on C with radius AB. Call the intersections D and E.
Segments CD and CE are both congruent to segment AB. Each is a valid answer.
NOTE: The measurements shown are only to show that the segments have the same measure and are congruent. They are not part of the construction!
Segment Congruent to a Segment Practice Worksheet
Angle Congruent to an Angle
Plan:
1)Draw a circle around A. The radius is unimportant. Call its intersections with A “B” and “C”.
2)Draw a circle around P with radius AB. Call its intersection with line l Q.
3)Measure BC with the compass and draw a circle around Q with radius BC.
4)Label the intersections of circles P and Q “R” and “S”.
Angles RPQ and QPS are both congruent to angle CAB.
Step 1: Draw a circle around A. Call its intersections with angle A “B” and “C”.
Step 2: Draw a circle around P with radius AB. Call its intersection with line l Q.
Step 3: Measure BC with the compass and draw a circle around Q with radius BC. Call the intersections with circle P “R” and “S”.
Angles RPQ and QPS are both congruent to angle CAB!
NOTE: The measurements shown are not part of the construction; they are there to test congruence.
Angle Congruent to an Angle Practice Worksheet
Perpendicular Bisector of a Segment
Plan:
1)Find two points equidistant from A and B
2)Draw a line between them; all points on the line will be equidistant from A and B
The line is a perpendicular bisector!
Step 1: Draw circles with radius AB centered on A and B.
Step 2: Draw line CD. This is the perpendicular bisector of segment AB!
NOTE: The same construction works to find the midpoint of a segment, since the midpoint is where the segment intersects its perpendicular bisector.
Perpendicular Bisector of a Segment Practice Worksheet
Construct a perpendicular bisector for each of the following segments.
Perpendicular through a Point
Plan:
1)Find two points on j which are equidistant from C.
2)Construct the perpendicular bisector for the segment joining the two points.
The perpendicular bisector will be perpendicular to J, and will pass through C since it’s equidistant from the two points.
Step 1) Draw a circle centered on C; make it big enough to intersect with line j twice.
Step 2: Construct the Perpendicular Bisector of line EF (see the Perpendicular Bisector construction).
The bisector goes through point C!
Perpendicular Through a Point Practice Worksheet
Parallel Through a Point
Plan:
- Draw a line through the point intersecting the line. This will be a parallel line transversal.
- Copy the angle made by the new line with the original up to the point.
This will be a corresponding angle, so we now have a parallel line.
Step 1: Draw a line connecting C to j. This will be the transversal.
We call the intersection D.
Call another point on the line E.
Step 2: Duplicate <CDE to corresponding <FCG.
Line CG is parallel to line j by the Corresponding Angles Converse.
Parallel Through a Point Worksheet
Construct a parallel line for each of the following segments that goes through its nearby point.
AngleBisector
Plan:
1)Draw a circle centered at the vertex of the triangle.
2)Draw a perpendicular bisector for the segment determined by the two points.
This is the angle bisector!
Step 1: Draw a circle centered at the vertex of the triangle; the radius should be small enough so that the circle intersects both adjacent sides of the triangle.
Construct a perpendicular bisector for segment DE. This is the angle bisector!
Angle Bisector Practice Worksheet
Construct a bisector for each of the following angles.
Circumscribe a Circle around a Triangle
Plan:
1)Construct perpendicular bisectors for the sides of the triangle. They meet at the circumcenter.
2)Draw a circle with center at the circumcenter, and radius going out to a corner of the triangle.
This circle will intersect all three vertices of the triangle, so it is the circumscribed circle.
Step 1: Construct perpendicular bisectors for the sides of the triangle. They meet at the circumcenter.
NOTE: We really only need two of the perpendicular bisectors, since we know that the third would meet them at the same intersection point.
Step 2: Draw a circle with center at J (the circumcenter) and extending out to vertex D.
This is the circumscribed circle!
NOTE! Either of the other vertices (E or F) would work just as well, since all vertices are the same distance from the circumcenter.
Circumscribe a Circle around a Triangle Practice Worksheet
Inscribe a Circle within a Triangle
Plan:
1)Construct the angle bisectors for the vertices of the triangle. They meet at the incenter.
2)Construct a perpendicular line segment from the incenter to any side of the triangle.
3)Draw a circle with center at the incenter, and radius extending out to the intersection of the perpendicular from (2) with the side.
Step 1: Construct the angle bisectors for the vertices of the triangle. They meet at the incenter.
NOTE: Two angle bisectors is enough, since they all intersect at the same point.
Step 2: Draw a perpendicular to side DF intersecting V.
Step 3: Draw a circle around V of radius VW. This is the inscribed circle!
Inscribe a Circle within a Triangle Practice Worksheet
Inscribe a circle within each of the following triangles.
Median of a Triangle
Plan:
1)Construct the midpoint of a side of the triangle.
2)Construct a line segment from the vertex opposite the side to the midpoint.
That’s the median!
Step 1: Construct the midpoint for a side of the triangle.
Step 3: Construct a segment from the midpoint to the opposite vertex.
That’s the median!
Median of a Triangle Practice Worksheet
Construct the median at vertex A of each of the following triangles.
Centroid of a Triangle
Plan:
1)Construct medians for the vertices of the triangle.
Their intersection is the centroid!
Step 1) Construct two medians of the triangle. (NOTE: We don’t need the third median, because it is concurrent with the other two.)
Their intersection is the centroid.
NOTE: The measurements are not part of the construction; they are shown to illustrate the 1:2 ratio into which the centroid divides each median.
Centroid of a Triangle Practice Worksheet
Construct the centroid of each of the following triangles.
Orthocenter of a Triangle
Plan:
1)Construct altitudes for each side of the triangle. Their intersection is the orthocenter.
Step 1: Construct altitudes for the sides of the triangle. (See the Perpendicular through a Point construction.) Two altitudes are enough, since the three are concurrent.
The intersection is the orthocenter.
Orthocenter of a Triangle Practice Worksheet
Construct the orthocenter of each of the following triangles.
Midsegment of a Triangle
Plan:
1)Construct midpoints for two sides of a triangle.
2)Draw a segment connecting the midpoints.
That’s the midsegment!
Step 1: Construct the midpoints of two sides. See the “Constructing a Perpendicular Bisector” construction.
Step 2: Draw a line segment connecting the two midpoints. This segment is the midsegment. Notice that it’s half the length of the third side of the triangle, and parallel to it.
NOTE: The measurements shown are not part of the construction. They are there to illustrate the ratio of two between the segment lengths, and (using corresponding angles of a transversal) that the midsegment parallels the third side.
Midsegment Practice Worksheet
Construct all midsegments of each of the following triangles.
Reflect a Point through a Line
Reflection Practice
Reflect each vertex of the triangle in the given line. Then, draw in the sides.
Rotate a Point Around a Point By an Angle
Rotation Practice
Rotate each vertex of the triangle around A by B, clockwise. Then, draw in the sides.
Glossary
Altitude / Line segment perpendicular to a side of a triangle, and going to the vertex opposite the sideAngle Bisector / A ray which starts at the vertex of an angle, and which divides the angle into two equal parts.
Centroid / Point of concurrency of a triangle’s medians
Circumcenter / The center of a circle circumscribing a triangle. Point of concurrency of the perpendicular bisectors of the triangle’s sides.
Circumscribe a Circle / To draw a circle around a triangle which intersects all three of its vertices. The circumscribed circle is the smallest circle containing the triangle.
Concurrent Lines / Lines intersecting at a point
Incenter / The center of a circle inscribed within a triangle. Point of concurrency of the angle bisectors of a triangle.
Inscribe a Circle / To draw a circle within a triangle that touches all three sides. The inscribed circle is the largest circle contained within the triangle.
Median / A line connecting a vertex of a triangle with the midpoint of the side opposite the vertex
Midsegment / Line segment connecting the midpoints of two sides of a triangle. Parallel to the third side, and half its length.
Orthocenter / Point of concurrency of a triangle’s altitudes
Perpendicular Bisector / A line which is perpendicular to a segment, and which divides the segment into two equal parts.
Point of Concurrency / The point at which concurrent lines intersect.