Geometric Construction Notes 2

Table of Contents

Construct a 45 Degree LineConstruct an Inscribed

Square From a SideConstruct a Circumscribed Square from a SideInscribed HexagonInscribed Octagon from Square

Divide Line Into Equal PartsConstruct an Arc Tangent to Two LinesConstruct an Arc Tangent to an Arc and a LineConstruct an Arc Tangent to Two Arcs

Construct a 45 degree line with the CompassBegin with a line

1. Bisect the line

Construct a 45 degree line with the Compass

2. Measure the length of ½ of the line from the bisector to the ep.

3. Without adjusting the compass radius, draw an arc intersecting the bisector

Construct a 45 degree line with the Compass

4. Connect the ep of the line to the intersection of the arc and bisector

Construct a 45 degree line with the Compass- Solution

Construct an Inscribed Square from a Side

Begin with a Side

1. Draw intersecting centerlines (CL) from the end points of the given diagonal – use 45 degree angle construction.

Construct an Inscribed Square from a Diagonal Side

2. Set the compass point on the intersection of the cls and adjust the radius of the compass to one of the eps of the diagonal.

3. Use the radius to draw a circle. Both eps of the line should touch the perimeter of the circle

Construct an Inscribed Square from a Diagonal Side

4. Connect the intersections of the cls and the circle perimeter to form the missing sides.

Construct a Circumscribed Square from a Side

Begin with a Side

1. Draw intersecting lines from the end points of the given diagonal – use 45 degree angle construction.

Construct a Circumscribed Square from a Side

2. Bisect the given line to find the radius of the circle. Your construction may overlap other constructions

Note: Extend the bisector on both sides of the cp

Construct a Circumscribed Square from a Side

3. Draw a line perpendicular to the bisector at the cp using the steps for drawing a perpendicular from a point on a line .

Construct a Circumscribed Square from a Side

4. Measure the radius of the circle by setting the point of the compass to the intersection of the two cls, the center point (cp), and the lead to the intersection of the bisect and given side

5. Draw a circle that intersects the diagonals and cls.

Construct a Circumscribed Square from a Side

6. Draw lines from the ep of the given line tangent to the circle at the intersection of the cls and circle perimeter to the diagonal

Construct a Circumscribed Square from a Side

7. Draw a fourth side by connecting the eps of the two sides drawn passing through the tangent point of the circle

Construct a Circumscribed Square from a Side- Solution

Construct an Inscribed Hexagon to a Given Circle

Begin with a circle and cp

1. Set the compass point at any point along the circle perimeter and measure the circle radius with the compass

Construct an Inscribed Hexagon to a Given Circle

2. Without adjusting the compass radius or moving the point draw and arc intersecting the perimeter of the circle.

3. Without adjusting the compass radius move the compass point to the intersection of the arc and circle and draw a second arc intersecting the circle

Construct an Inscribed Hexagon to a Given Circle

4. Repeat step 3 until you have 5 arcs intersecting the circle perimeter. These along with the original point are the vertices of the hexagon.

Construct an Inscribed Hexagon to a Given Circle

5. Connect the vertices with a straight edge to create the sides

Construct an Inscribed Octagon to a Given Square

Begin with a square

1. Find the cp of the square by connecting opposite corners of the square

Construct an Inscribed Octagon to a Given Square

3. Use the compass to measure the distance from one vertices of the square and the cp

4. Without adjusting the compass radius draw arcs from each vertices intersecting the adjacent sides

Construct an Inscribed Octagon to a Given Square

5. Connect intersections of the arcs and perimeter of the square

Construct an Inscribed Octagon to a Given Square- Solution

Divide a Line Into Equal Parts

Begin with a line

1. Draw a diagonal line from one ep of the line

2. Using the compass draw arcs intersecting the angle to create equal spacing.

Note: The number of spaces is determined by the number of divisions

Divide a Line Into Equal Parts

3. Using the compass measure the distance between the ep of the line and last intersection along the diagonal.

4. Without adjusting the compass radius draw an arc from the opposite ep.

Divide a Line Into Equal Parts

5. Using the compass measure the length of the diagonal to the last intersection.

6. Without adjusting the compass radius draw an arc from the opposite ep. that intersects the other arc

Divide a Line Into Equal Parts

7. Use a straightedge to connect the ep of the line and the intersection. This line is parallel to the diagonal.

Divide a Line Into Equal Parts

8. Using the same radius as the first set of intersecting arcs draw an identical set.

Divide a Line Into Equal Parts

9. Connect the intersections of the arcs and diagonals with parallel lines.

The parallel lines divide the given line into equal parts where they intersect the given line.

Divide a Line Into Equal Parts- Solution

Construct an Arc Tangent to Two Lines

Begin with any angle or pair of non-parallel lines

1. Use the steps to draw a parallel line through a point. The point must be the same distance away from the line as the radius of the arc Note: You may use a scale to locate

the point

Construct an Arc Tangent to Two Lines

2. Use the compass to measure from the intersection of the lines, the cp, to the intersection of one of the lines and a given line, the radius.

3. Draw the arc between both lines.

Construct an Arc Tangent to Two Lines- Solution

Construct an Arc Tangent to an Arc and a Line

Begin with the cp of an arc and a given line

1. Use the steps to draw a parallel line to the given line the same distance away as the radius of the tangent arc

Note: You may use a scale to locate the point

Construct an Arc Tangent to an Arc and a Line

2. 2. Draw the circle or arc to the given radius or diameter

Construct an Arc Tangent to an Arc and a Line

3. Draw a concentric arc with a radius equal to the radius of the given arc or circle plus the radius of the tangent arc

Construct an Arc Tangent to an Arc and a Line

4.Using the steps to draw a perpendicular through a point on a line draw a center line at the intersection of the arc and line

Construct an Arc Tangent to an Arc and a Line

5.Using the steps to draw a perpendicular through a point on a line draw a center line at the intersection of the arc and line

Construct an Arc Tangent to an Arc and a Line

6. Set the compass point to the intersection of the line and arc, the cp, and measure the radius to the intersection of the given line and perpendicular

7. Draw the arc tangent to the circle and line

Construct an Arc Tangent to an Arc and a Line- Solution

Construct an Arc Tangent to Two Arcs

Begin with the cps of two arcs

1. Draw the circles or arcs to the given radii or diameters

Construct an Arc Tangent to Two Arcs

2. Draw intersecting concentric arcs with a radii equal to the radius of each of the given arcs or circles plus the radius of the tangent arc

Construct an Arc Tangent to Two Arcs

3. Draw a line from the cp of one of the circles or arcs and the intersection of the arcs

Construct an Arc Tangent to Two Arcs

4. Place the compass point on the intersection of the arcs and the intersection of the circle and line.

5. Draw the tangent arc.

Construct an Arc Tangent to Two Arcs- Solution