Constructions

 

Step-by-step instructions are provided for constructing (with straight-edge and compass) each of the following:

 

  • A circle with a given radius
  • A congruent segment
  • A congruent angle
  • A congruent triangle
  • Parallel lines
  • An angle bisector
  • A perpendicular bisector
  • A perpendicular line from a point off the line
  • A perpendicular line through a point on the line
  • The perpendicular bisector of a segment
  • An altitude of a triangle

 

 

 

A Circle with a Given Radius

 

  1. Before starting this construction, you should be given a segment which is to be the length of your radius.
  2. Draw a point on your paper which will be the center of your circle.
  3. Open up your compass so that its opening is the length of the given segment.
  4. Without letting the compass slip, place the point of your compass on your previously drawn center and draw your circle.
  5. You should now have a circle whose radius is congruent to the given segment.

 

 

A Congruent Segment

 

  1. For this construction, you are given a segment and you wish to construct a second segment which is congruent to the first.
  2. Draw a ray on your paper.
  3. Open up your compass so that its opening is the length of the given segment.
  4. Without letting the compass slip, place the point of your compass on the endpoint of the ray.
  5. Mark a small arc on the ray with the compass.
  6. Draw a point where the ray and arc intersect.
  7. The congruent segment begins at the ray’s endpoint and ends at this point of intersection.

 

 

A Congruent Angle

 

  1. For this construction, you are given an angle and you wish to construct a second angle congruent to the first.
  2. Draw a ray on your paper.
  3. Place the point of your compass on the vertex of the given angle.
  4. Draw an arc which intersects both rays of the given angle.
  5. Without letting the compass slip, place the point of your compass on the endpoint of the previously drawn ray and draw the same arc.
  6. Go back to the given angle.  Open up you compass so that is opening extends from where the arc intersects one side of the angle to where the arc intersects the other side of the angle.
  7. Go to your previously drawn ray.  Without letting the compass slip, place the point of your compass on the point where the arc and ray intersect.
  8. Draw an arc that intersects the first arc.
  9. Draw a ray beginning at the endpoint of the original ray and extending through the point of intersection of the two arcs.
  10. You should no have an angle that is congruent to the given angle.

 

 

A Congruent Triangle

 

  1. For this construction, you are given a triangle and asked to construct a second triangle that is congruent to the first.  There are several ways to do this.  The steps below use the idea of SSS congruency.
  2. Draw a ray on your paper.
  3. Open up your compass so that its opening is the length of the first side of your triangle.
  4. Without letting the compass slip, place the point of your compass on the endpoint of the ray.
  5. Mark a small arc on the ray with the compass.
  6. Draw a point where the ray and arc intersect.
  7. The segment which begins at the ray’s endpoint and ends at this point of intersection is the first side of your triangle.
  8. Open up your compass so that its opening is the length of the second side of your triangle.
  9. Without letting the compass slip, place its point on one endpoint of the first side of your soon-to-be congruent triangle.
  10. Draw an arc.
  11. Open up your compass so that its opening is the length of the third side of your triangle.
  12. Without letting the compass slip, place its point on the other endpoint of the first side of your soon-to-be congruent triangle.
  13. Draw an arc.
  14. Draw a point where the two arcs intersect.  This point is the third point of your triangle.
  15. Draw segments connecting this third point to the endpoints of your first side.
  16. You should now have a triangle which is congruent to the original triangle.

 

 

Parallel Lines

 

  1. Given a line, you are being asked to construct a second line which is parallel to this first line.
  2. Draw a point somewhere above the given line.
  3. Draw a transversal that intersects both the point and the given line.
  4. Place the point of your compass at the point of intersection of the transversal and the given line.
  5. Draw an arc that intersects both the given line and the transversal.
  6. Without letting the compass slip, place the point of your compass on the point on the transversal that you previously drew.
  7. Draw the same arc.
  8. Go back to the angle formed by the transversal and the given line.
  9. Open up your compass so that its opening extends from where the arc intersects the given line to where the arc intersects the transversal.
  10. Go to your previously drawn point above the line.
  11. Without letting the compass slip, place the point of your compass on the point where the arch and transversal intersect.
  12. Draw a second arc that intersects this first arc.
  13. Draw a line that passes through the original point on the transversal and extending through the point of intersection of the two arcs.
  14. You should now have a line that is parallel to the first line.

 

 

An Angle Bisector

 

  1. In this construction, you are given an angle and your goal is to draw a ray which bisects the angle.
  2. Begin by placing the point of your compass on the vertex of the angle.
  3. Draw an arc that intersects both sides of the angle.
  4. Without letting the compass slip, place the point of your compass on one of the points of intersection.
  5. Draw an arc in the interior of the angle.
  6. Again without letting the compass slip, place the point of your compass on the other point of intersection.
  7. Draw an arc in the interior of the angle which intersects the other arc.
  8. Draw a ray that begins at the vertex of the angle and extends through the point of intersection of the two arcs.
  9. You should now have a ray which bisects the given angle.

 

 

 

A Perpendicular Line from a Point Off the Line

 

  1. Given a line and a point not on that line, your goal in this construction is to construct a perpendicular line to the given line that passes through the given point.
  2. Place the point of your compass on the given point off the line.
  3. Open up the compass so that you are able to draw an arc that intersects the given line in two places.  Draw this arc.
  4. Without letting the compass slip, place the point of the compass on one of the points of intersection.
  5. Draw an arc on the opposite side of the line from the given point.
  6. Again, without letting the compass slip, place the point of the compass on the other point of intersection.
  7. Draw an arc on the opposite side of the line from the given point.  This arc should intersect the last arc.
  8. Draw a point at the point of intersection of these last two arcs.
  9. Draw a line that passes through this point of intersection and the given point.
  10. This line should be perpendicular to the original line.

 

 

A Perpendicular Line through a Point on the Line

 

  1. Given a line and a point on the line, your goal in this construction is to construct a perpendicular line to the given line that passes through the given point on that line.
  2. Place the point of your compass on the given point on the line.
  3. Open up the compass and draw an arc that intersects the given line in two places.
  4. Open up the compass a little wider.
  5. Place the point of the compass on one of the points of intersection.
  6. Draw an arc on both sides of the line.
  7. Without letting the compass slip, place the point of the compass on the other point of intersection.
  8. Draw an arc on both sides of the line.  This arc should intersect the last arc.
  9. Draw a point at the point of intersection of these last two arcs on each side of the given line.
  10. Draw a line that passes through these points of intersection and the given point.
  11. This line should be perpendicular to the original line.

 

 

The Perpendicular Bisector of a Segment

 

 

  1. The purpose of this construction is to construct a perpendicular bisector of a given segment.
  2. Open up your compass so that the opening is about ¾ as long as the given segment.
  3. Without letting the compass slip, place the point of your compass on one endpoint of the given segment.
  4. Draw an arc above and below the given segment.
  5. Again without letting the compass slip, place the point of your compass on the other endpoint of the given segment.
  6. Draw an arc above and below the given segment.
  7. These arcs should intersect.  Draw the points of intersection.
  8. Draw a line that passes through these two points of intersection.
  9. This line should be perpendicular to the original segment and it should bisect the original segment.

 

 

An Altitude of a Triangle

 

  1. For any given triangle, there are three altitudes that may be constructed, one from each vertex.
  2. Choose the vertex from which you wish to draw the altitude.
  3. Extend the opposite side in the triangle, if necessary.
  4. You are being asked to construct a perpendicular to the given line (the opposite side of the triangle) through a point off that line (the selected vertex).  See the instructions for constructing a perpendicular line through a point no on the line.