mr - loudoun county public schools / overvie · web viewsegment congruent to a segment practice...

7 May 2023 Mr. Feist’s Geometric Construction Cookbook of 62

Mr. Feist’s

Mr. Bruce FeistDecember, 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.

7 May 2023 Mr. Feist’s Geometric Construction Cookbook Page 3 of 62

Contents

Segment Congruent to a Segment................................................................................................................................................................4Segment Congruent to a Segment Practice Worksheet...........................................................................................................................6

Angle Congruent to an Angle......................................................................................................................................................................7Angle Congruent to an Angle Practice Worksheet................................................................................................................................12

Perpendicular Bisector of a Segment.........................................................................................................................................................12Perpendicular Bisector of a Segment Practice Worksheet....................................................................................................................16

Perpendicular through a Point...................................................................................................................................................................17Perpendicular Through a Point Practice Worksheet..............................................................................................................................20

Parallel Through a Point............................................................................................................................................................................21Parallel Through a Point Worksheet......................................................................................................................................................24

Angle Bisector...........................................................................................................................................................................................25Angle Bisector Practice Worksheet.......................................................................................................................................................28

Circumscribe a Circle around a Triangle...................................................................................................................................................29Circumscribe a Circle around a Triangle Practice Worksheet...............................................................................................................32

Inscribe a Circle within a Triangle............................................................................................................................................................33Inscribe a Circle within a Triangle Practice Worksheet........................................................................................................................36

Median of a Triangle.................................................................................................................................................................................37Median of a Triangle Practice Worksheet.............................................................................................................................................41

Centroid of a Triangle................................................................................................................................................................................42Centroid of a Triangle Practice Worksheet...........................................................................................................................................45

Orthocenter of a Triangle...........................................................................................................................................................................46Orthocenter of a Triangle Practice Worksheet......................................................................................................................................48

Midsegment of a Triangle..........................................................................................................................................................................49Midsegment Practice Worksheet...........................................................................................................................................................52

Reflect a Point through a Line...................................................................................................................................................................53Reflection Practice.................................................................................................................................................................................56

Rotate a Point Around a Point By an Angle..............................................................................................................................................57Rotation Practice....................................................................................................................................................................................60

Glossary.....................................................................................................................................................................................................61


Segment Congruent to a Segment

Given: Segment ABGoal: 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.

A B

C


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!

AB = 2.46 cm

CD = 2.46 cmEC = 2.46 cm

E

D

A B

C


Segment Congruent to a Segment Practice Worksheet

Construct a segment congruent to each of the following.

A B C

D

E

F


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.

l

A

P


Step 1: Draw a circle around A. Call its intersections with angle A “B” and “C”.

l

C

A

P

B


Step 2: Draw a circle around P with radius AB. Call its intersection with line l Q.

lQ

C

A

P

B


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”.

l

S

R

Q

C

A

P

B


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.

l

m QPS = 28.76

m RPQ = 28.76

m CAB = 28.76

S

R

Q

C

A

P

B


Angle Congruent to an Angle Practice Worksheet

Perpendicular Bisector of a Segment

Construct an angle congruent to each of the below.

B

A

C

D

E F

G

H

I


Plan:1) Find two points equidistant from A and B2) Draw a line between them; all points on the line will be equidistant from A and B

The line is a perpendicular bisector!

A B


Step 1: Draw circles with radius AB centered on A and B.

D

C

A 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.

D

C

A B


Perpendicular Bisector of a Segment Practice Worksheet

Construct a perpendicular bisector for each of the following segments.

A B C

D

E

F


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.

j

C


Step 1) Draw a circle centered on C; make it big enough to intersect with line j twice.

j F

C

E


Step 2: Construct the Perpendicular Bisector of line EF (see the Perpendicular Bisector construction).The bisector goes through point C!

j I

H

G

F

C

E


Perpendicular Through a Point Practice Worksheet

Construct a perpendicular for each of the following segments that goes through its nearby point.

A B C

D

E

F

H

IG


Parallel Through a Point

j

C

Plan:1. Draw a line through the point intersecting the line. This will be a parallel line transversal.2. 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.


j

C

D E

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.


j

GC

D E

F

Step 2: Duplicate <CDE to corresponding <FCG.Line CG is parallel to line j by the Corresponding Angles Converse.


Parallel Through a Point WorksheetConstruct a parallel line for each of the following segments that goes through its nearby point.

A B C

D

E

F

H

IG


Angle Bisector

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!

j

kC


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.

D

C E


Construct a perpendicular bisector for segment DE. This is the angle bisector!

H

D

C E


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.

D

E

F


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.

JD

E

F


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.

JD

E

F


Circumscribe a Circle around a Triangle Practice Worksheet

Circumscribe a circle around each of the following triangles.


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.

D

E

F


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.

VD

E

F


W

V

D

E

F

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

W

V

D

E

F


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!

A

B

C


Step 1: Construct the midpoint for a side of the triangle.

D

A

B

C


Step 3: Construct a segment from the midpoint to the opposite vertex.

That’s the median!

D

A

B

C


Median of a Triangle Practice Worksheet

Construct the median at vertex A of each of the following triangles.

A A

A


Centroid of a Triangle

Plan:1) Construct medians for the vertices of the triangle.

Their intersection is the centroid!

A

B

C


Step 1) Construct two medians of the triangle. (NOTE: We don’t need the third median, because it is concurrent with the other two.)

ED

A

B

C


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.

3.14 cm

1.57 cm

3.36 cm

1.68 cm

F

ED

A

B

C


Centroid of a Triangle Practice WorksheetConstruct 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.

A

B

C


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.

H1

A

B

C


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!

A

B

C


A

B

C

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.

mACB = 51.30

mMNB = 51.30

m AC = 5.16 cm

m MN = 2.58 cm

NM

A

B

C


Midsegment Practice Worksheet

Construct all midsegments of each of the following triangles.


Reflect a Point through a Line

m

Concept: 1) Draw a perpendicular to m going through A. (See the "perpendicular through a point" construction.)Call this perpendicular "n". Call the intersection of m with n "B".2) Measure AB.Draw a circle around B with radius AB. It intersects with line AB at A (the preimage) and a new point, C (the image)

A


n

m

Draw a perpendicular to m going through A. (See the "perpendicular through a point" construction.)Call this perpendicular "n". Call the intersection of m with n "B".

B

A


n

m

Measure AB.Draw a circle around B with radius AB. It intersects with line AB at A (the preimage) and a new point, C (the image)C

B

A


Reflection PracticeReflect each vertex of the triangle in the given line. Then, draw in the sides.


Rotate a Point Around a Point By an Angle

Concept: 1) Draw a ray from M (the center of the rotation) through A.2) Make an angle with one side ray MA, and congruent to B. (See the construction for copying anangle).3) Draw a circle around M with radius AM. It will intersect the angle at the image of A.

A

MB


Draw a ray from M (the center of the rotation) through A.

A

MB


36.90°

36.90°

36.90°

2.44 cm

2.44 cm

Make an angle with one side MA, and congruent to B. (See the construction for copying an angle).We can most easily do this as follows:Draw a circle of radius AM around B. Call its intersections with B "C" and "D".Draw a circle of radius AM around M.Measure CD with the compass.Draw a circle of radius CD around A.This will intersect the circle around M in two points. One is the image if we rotate clockwise, the other isthe image if we rotate counterclockwise. See the way that the angle measurements are the same?

A'

A'

D

C

A

MB


Rotation PracticeRotate each vertex of the triangle around A by B, clockwise. Then, draw in the sides.

A

B


GlossaryAltitude 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 mediansCircumcenter 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 pointIncenter 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 vertexMidsegment 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 altitudesPerpendicular 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.

mr - loudoun county public schools / overvie · web viewsegment congruent to a segment practice...

Documents