Geometry construction of square

This section of the article shows how to construct (or draw) a square with the length of a side given.

It begins with a given line segment AB. It then constructs a perpendicular at one end of the line, which will turn into the second side of the square. The compass is then set to the length of the given side, and the other three sides are marked off.

Step-by-step instructions On How To carry out the construction operation

This step by step instruction can be handy for your use for a lesson note or when you are not in the presence of a computer.

Proof

The case and argument Explanation behind the argument

1. Angle ABC is 90° (a right angle) By construction. See Constructing a perpendicular to a line at a point for the proof.

2. Segments BC, CD and DA are the same length as AB They were all drawn with the same compass width – AB

3. Thus ABCD is a square. This is because it is made up of four congruent sides and all interior angles are 90°

- Q.E.D

The instructions for the Construction of a square

After you have done the following Your work ought to appear like this

We begin with a given line segment AB> This will become one side of the square. 

Note: Steps 1 through 5 construct a perpendicular to line AB at the point B. This is the same construction as Constructing the perpendicular at a point on a line

1. Enlarge the line AB to the right. 

2. Set the compasses on B and any suitable width. Scribe an arc on each side of B, prodcing the two points F and G. 

3. With the compasses on G and any convenient width, draw an arc above the point B. 

4. Without altering the compasses' width, place the compasses on F and draw an arc above B, crossing the preceding arc, and creating point H 

5. Draw a line from B through H.

This line is perpendicular to AB, so the angle ABH is a right angle (90°);

This will turn into the second side of the square 

We now construct four sides of the square the same length as AB

6. Put the compasses on A and set its width to AB. This width will be held unchanged as we create the square's other three sides. 

7. Draw an arc above point A. 

8. Without altering the width, move the compasses to point B. Draw an arc across BH producing point C - a vertex of the square. 

9. Without altering the width, move the compasses to C. Draw an arc to the left of C across the exiting arc, producing point D - a vertex of the square. 

10. Draw the lines CD and AD 

Done. ABCD is a square where each side has a length AB

How to construct a square inscribed in a given circle. The construction goes on as follows:

1. A diameter of the circle is drawn.

2. A perpendicular bisector of the diameter is drawn with the use of the method described in Perpendicular bisector of a segment. This is as well a diameter of the circle.

3. The resulting four points on the circle are the vertices of the inscribed square.

No center point?

If the circle's center point is not given, it can be constructed with the use of the method in Constructing the center of a circle.

Printable step-by-step instructions

The instruction is provided in the table below in a step by step printable instruction sheet, which can be used for making handouts or when a computer is not handy.

Proof

- Q.E.D

Argument Reason

1. AC is a diameter of the circle O A diameter is a line through the center of a circle.

2. BD is a diameter of the circle O It was drawn using the method in Perpendicular bisector of a line. See that article for proof. The center of a circle bisects the diameter, thus BD passes through the center.

3. AC, BD are perpendicular BD was drawn using the method in Perpendicular bisector of a line. See that article for proof.

4. AC, BD bisect each other Both are diameters of the circle O. (1), (2) and the center of a circle bisects its diameter.

5. ABCD is a square Diagonals of a square bisect each other at 90°. (3), (4)

6. ABCD is an inscribed square All vertices lie on the given circle O

Step by step instructions

After you have carried out the following steps Your work ought to appear as follows

We begin with a given line segment AB> This will become one side of the square. 

Note: Steps 1 through 5 construct a perpendicular to line AB at the point B. This is the same construction as Constructing the perpendicular at a point on a line

1. Extend the line AB to the right. 

2. Set the compasses on B and any suitable width. Scribe an arc on each side of B, forming the two points F and G. 

3. With the compasses on G and any suitable width, draw an arc above the point B. 

4. Without altering the compasses' width, place the compasses on F and draw an arc above B, passing the preceding arc, and producing point H 

5. Draw a line from B through H.

This line is perpendicular to AB, in such a way that the angle ABH is a right angle (90°);

This will turn into the second side of the square 

We at this point construct four sides of the square the same length as AB

6. Set the compasses on A and set its width to AB. This width will be held the same as we construct the square's other three sides. 

7. Draw an arc on top of point A. 

8. Without altering the width, move the compasses to point B. Draw an arc across BH producing point C - a vertex of the square. 

9. Without altering the width, move the compasses to C. Draw an arc to the left of C across the exiting arc, producing point D - a vertex of the square. 

10. Draw the lines CD and AD 

Done. ABCD is a square where each side has a length AB

This is the step-by-step, printable instruction of a square inscribed in a circle

After you have done the following The result you would obtain would look like this

Begin with the given circle, center O. 

If the circle center point is not given, you can construct the center using the method shown in Finding the center of a circle. 

1 Mark a point A on the circle. This will become one of the vertices of the square. 

2 Draw a diameter line from the point A, through the center and continue across the circle to create point C. 

3 Set the compass on A and set the width to a little more than the distance to O. 

4 Draw an arc over and below O. 

5 Move the compass to C and repeat. 

6 Draw a line through where the arc pairs cross, making it long enough to touch the circle at top and bottom, producing the new points B and D. 

This is a diameter at right angles to the first one AC. 

7 Draw a line between each successive pairs of points A, B, C, D 

At this stage, you are done. ABCD is a square inscribed in the given circle. 

Geometry construction of Regular hexagon, given one side with the use of a compass and straightedge

How to construct a regular hexagon with one side given

The construction begins by finding the center of the hexagon, then drawing its circumcircle, which is the circle that passes through each vertex. The compass then goes around the circle marking off each side.

Printable step-by-step instruction that can be used for making handouts or when a computer is not available

Explanation of method

This construction is very similar to Constructing a hexagon inscribed in a circle, except we are not given the circle, but rather one of the sides. Steps 1-3 are there to draw this circle, and from then on the constructions are the same.

The center of the circle is obtained using the fact that the radius of a regular hexagon (distance from the center to a vertex) is equal to the length of each side.

The picture below shows how the final drawing will look like.

The case Explanation

1 ABCDEF is a hexagon It is a polygon with six sides.

2 AB, BC, CD, DE, EF, FA are all congruent. Drawn with the same compass width AF.

3 A, B, C, D, E, F all lie on the circle O By construction

4 ABCDEF is a regular hexagon From (1), (2). All its vertices lie on a circle, and all sides are congruent. This defines a regular hexagon.

- Q.E.D

Construction of a hexagon given one side

This is the step-by-step, printable instruction to guide you through.

After doing this Your work should look like this

We begin with a line segment AF. This will become one side of the hexagon. Due to the fact that we are constructing a regular hexagon, the other five sides will have the same length as well. 

1. Put the compasses' point on A, and set its width to F. the compasses ought to remain at this width for the rest of the construction. 

2. From points A and F, draw two arcs so that they interconnect. Mark this as point O. 

This is the center of the hexagon's circumcircle.

3. Move the compasses to O and draw a circle. 

This is the hexagon's circumcircle - the circle that passes through all six vertices 

4. Shift the compasses on to A and draw an arc across the circle. This is the next vertex of the hexagon. 

5. Shift the compasses to this arc and draw an arc across the circle to create the next vertex. 

6. Move on in this way until you have all six vertices. (Four new ones plus the points A and F you began with.) 

7. Draw a line between each successive pairs of vertices. 

8. Done. These lines form a regular hexagon where each side is equal in length to AF. 

Geometry construction of Hexagon inscribed in a circle with the use of a compass and straightedge

This section shows how to construct (draw) a regular hexagon inscribed in a circle with a compass and straightedge or ruler. This is the largest hexagon that will fit in the circle, with each vertex touching the circle. In a regular hexagon, the side length is equal to the distance from the center to a vertex, therefore we make use of this fact to set the compass to the proper side length, then move around the circle marking off the vertices.

Printable step-by-step instructions

The above construction is available as a printable step-by-step instruction manual, which can be used for making handouts or in an absence of a computer.

Explanation of method

From the Definition of a Hexagon, each side of a regular hexagon is equal to the distance from the center to any vertex. This construction merely sets the compass width to that radius, and then moves that length off around the circle to produce the six vertices of the hexagon.

Proof

The picture below shows the final drawing and how it would look like:

The case Explanation for the case

1 A,B,C,D,E,F all lie on the circle center O By construction.

2 AB = BC = CD = DE = EF They were all drawn with the same compass width.

From (2) we see that five sides are equal in length, but the last side FA was not drawn with the compasses. It was the "left over" space as we stepped around the circle and stopped at F. So we have to prove it is congruent with the other five sides.

3 OAB is an equilateral triangle AB was drawn with compass width set to OA, 

and OA = OB (both radii of the circle).

4 m∠AOB = 60° All interior angles of an equilateral triangle are 60°.

5 m∠AOF = 60° As in (4) m∠BOC, m∠COD, m∠DOE, m∠EOF are all and 60 degrees;

due to the fact that all the central angles add to 360°,

m∠AOF = 360 - 5(60)

6 Triangle BOA, AOF are congruent SAS See Test for congruence, side-angle-side.

7 AF = AB CPCTC - Corresponding Parts of Congruent Triangles are Congruent

Therefore, at this point we have all the pieces to prove the construction

8 ABCDEF is a regular hexagon inscribed in the given circle • From (1), all vertices lie on the circle

• From (20), (7), all sides are the same length

• The polygon has six sides.

- Q.E.D View more on psalmfresh.blogspot.com for more educating and tricks article Thank you