# Construction (Geometry)

See how to construct a line segment/angle

by using a compass and a ruler.

6 examples and their solutions.

## Construction: Midpoint of a Line Segment

### Example

Construct the midpoint of the given line segment.

Solution Place the point of the compass at 1.

Draw two arcs

on the top and bottom of the segment.

Set the radius of the compass

that seems to be larger than

one half the length of the segment.

Draw two arcs

on the top and bottom of the segment.

Set the radius of the compass

that seems to be larger than

one half the length of the segment.

↓

Without changing the radius of the compass,

place the point of the compass at 2.

place the point of the compass at 2.

↓

Draw two arcs that pass through 3 and 4.

↓

Draw a line that passes through 3 and 4.

↓

The brown point is the midpoint

of the line segment.

of the line segment.

And the blue line is the perpendicular bisector

of the line segment.

of the line segment.

Close

## Construction: Congruent Angle

### Example

Construct an angle on the right ray

that is congruent to the left angle.

Solution that is congruent to the left angle.

Place the point of the compass at 1.

Draw an arc that passes through both rays.

Draw an arc that passes through both rays.

↓

Without changing the radius of the compass,

place the point of the compass at 2.

place the point of the compass at 2.

↓

Draw an arc (blue arc)

just like the previous arc.

just like the previous arc.

↓

Place the point of the compass at 3.

Set the radius of the compass 3 ~ 4.

Set the radius of the compass 3 ~ 4.

↓

Without changing the radius of the compass,

place the point of the compass at 5.

place the point of the compass at 5.

↓

Draw an arc that passes through 6.

↓

Draw a ray that passes through 6.

↓

Then the right side angle

is congruent to the left side angle.

is congruent to the left side angle.

Close

## Construction: Angle Bisector

### Example

Construct an angle bisector of the given angle.

Solution Place the point of the compass at 1.

Draw an arc that passes through both rays.

Draw an arc that passes through both rays.

↓

Place the point of the compass at 2.

Draw an arc

that seems to pass the middle of the angle.

Draw an arc

that seems to pass the middle of the angle.

↓

Without changing the radius of the compass,

place the point of the compass at 3.

place the point of the compass at 3.

↓

Draw an arc that passes through 4.

↓

Draw a ray that starts at 1

and that passes through 4.

and that passes through 4.

↓

Then the blue ray

is the angle bisector of the given angle.

is the angle bisector of the given angle.

Close

## Construction: Perpendicular Line from a Point on the Line

### Example

Construct a perpendicular line

that passes through the point on the line.

Solution that passes through the point on the line.

Place the point of the compass at 1.

Draw two arcs

that intersect with the given line.

Draw two arcs

that intersect with the given line.

↓

Place the point of the compass at 2.

Draw two arcs

on the top and bottom of the given point.

Draw two arcs

on the top and bottom of the given point.

↓

Without changing the radius of the compass,

place the point of the compass at 3.

place the point of the compass at 3.

↓

Draw two arcs that pass through 4 and 5.

↓

Draw a line that passes through 4 and 5.

↓

The blue line is the perpendicular line

that passes through the point on the given line.

that passes through the point on the given line.

Close

## Construction: Perpendicular Line from a Point Not on the Line

### Example

Construct a perpendicular line

that passes through the point not on the line.

Solution that passes through the point not on the line.

Place the point of the compass at 1.

Draw two arcs

that intersect with the given line.

Draw two arcs

that intersect with the given line.

↓

Place the point of the compass at 2.

Draw an arc

that is below the given point.

Draw an arc

that is below the given point.

↓

Without changing the radius of the compass,

place the point of the compass at 3.

place the point of the compass at 3.

↓

Draw an arc that passes through 4.

↓

Draw a line that passes through 1 and 4.

↓

The blue line is the perpendicular line

that passes through the point

not on the given line.

that passes through the point

not on the given line.

Close

## Constuction: Parallel Line

### Example

Construct a line

that is parallel to the given line

and that passes through the given point not on the line.

Solution that is parallel to the given line

and that passes through the given point not on the line.

Draw a line

that passes through

the given line and the given point 1.

that passes through

the given line and the given point 1.

↓

Place the point of the compass at 2.

Draw an arc that passes through 3 and 4.

Draw an arc that passes through 3 and 4.

↓

Without changing the radius of the compass,

place the point of the compass at 1.

place the point of the compass at 1.

↓

Draw an arc

just like the previous arc.

just like the previous arc.

↓

Place the point of the compass at 3.

Set the radius of the compass 3 ~ 4.

Set the radius of the compass 3 ~ 4.

↓

Without changing the radius of the compass,

place the point of the compass at 5.

place the point of the compass at 5.

↓

Draw an arc that passes through 6.

↓

Draw a line that passes through 1 and 6.

↓

Then the blue line is a parallel line

that passes through the given point.

that passes through the given point.

Close