Constructions
Some Euclidean Constructions for my Geometry class to do at home
Table of Contents
- Introduction to the Tools
- The Tools
- Line Segment Constructions
- Copy a Line Segment
- Add and Subtract Line Segments
- Create a Perpendicular Bisector/Bisect a Line Segment
- Create a Perpendicular Line Through a Point on the Line
- Create a Perpendicular Line Through a Point NOT on the Line
- Create a Parallel Line Through a Point
- Create a Square from One Side
- Angle Constructions
- Copy an Angle
- Add Angles
- Subtract Angles
- Create a Parallel Line Using Congruent Angles
- Bisect an Angle
- Create a 45 Degree Angle
- Create a 60 Degree Angle
- Create a 30 Degree Angle
- Triangle Constructions
- Copy a Triangle
- Create an Equilateral Triangle
- Create an Isosceles Triangle
- Inscribe a Circle Inside of a Triangle
- Circumscribe a Circle About a Triangle
- Circle Constructions
- Find the Center of a Circle Using Two Chords
- Create a Circle Touching Three Given Points
- Create a Tangent to a Circle Through a Given Point on the Circle
- Create a Tangent to a Circle Through a Given Point Outside the Circle
- Create a Square Inscribed in a Circle
- Create a Hexagon Inscribed in a Circle
- Create an Octagon Inscribed in a Circle
- EXTRA CREDIT: Create a Dodecagon Inscribed in a Circle
The Tools
Point Tool - [icon]/images/ggb/toolbar/mode_point.png[/icon] - Creates a new point when you click on the applet[br][br]Intersect Tool - [icon]/images/ggb/toolbar/mode_intersect.png[/icon] - Creates a point at the intersection of two objects. This could be done by clicking where the two objects intersect or by clicking one of the objects and then the other one. The second method will create points every time the two objects intersect (for example, if a line and a circle intersect twice, two points will be created).[br][br]Line Tool - [icon]/images/ggb/toolbar/mode_join.png[/icon] - Creates a line which goes on forever. This can be done by clicking two points or by clicking a place where there are no points and creating a point there.[br][br]Segment Tool - [icon]/images/ggb/toolbar/mode_segment.png[/icon] - Creates a line segment. This can be done by clicking two points or by clicking a place were there are no points and creating a point there.[br][br]Ray Tool - [icon]/images/ggb/toolbar/mode_ray.png[/icon] - Creates a ray. This can be done by clicking two points or by clicking a place were there are no points and creating a point there. The first point you select will always be the endpoint of the ray.[br][br]Polygon Tool - [icon]/images/ggb/toolbar/mode_polygon.png[/icon] - Creates a polygon by clicking in order the points you want to be the vertices of the polygon. You must click the first point again last to close the polygon.[br][br]Circle with Center through Point Tool - [icon]/images/ggb/toolbar/mode_circle2.png[/icon] - Creates a circle with a center at the point you click. Click again to set how large the circle is. [br][br]Compass Tool - [icon]/images/ggb/toolbar/mode_compasses.png[/icon] - Creates a circle. Click two points to set the radius of the circle, and then a point to be the center (could be one of the first two points). You could also click a line segment to set the radius.[br][br]Angle Measure Tool - [icon]/images/ggb/toolbar/mode_angle.png[/icon] - Finds the angle measure in degrees. Click a point on one side of the angle, then the vertex, then a point on the other side of the angle. Clicking counterclockwise will give you the interior angle. Clicking clockwise will give you the measure around the outside of the angle. If it doesn't measure the way you want, undo it and click the points in the opposite order.[br][br]Distance Tool - [icon]/images/ggb/toolbar/mode_distance.png[/icon] - Finds the distance between two points or the length of a line segment. Click the two points or click the line segment.[br][br]Move Graphics View Tool - [icon]/images/ggb/toolbar/mode_translateview.png[/icon] - Lets you move the screen around so you can see things that may have been moved or created outside of the visual window.[br][br]Zoom In Tool - [icon]/images/ggb/toolbar/mode_zoomin.png[/icon] - Let's you zoom in. Objects will not become larger, but the spaces between them will be. Sometimes this is helpful when two points end up very close to each other so you can select one of them.[br][br]Zoom Out Tool - [icon]/images/ggb/toolbar/mode_zoomout.png[/icon] - Let's you zoom out. This is helpful if you've created a circle and need to see the whole thing but parts of it are outside of the visual window.[br][br]Show/Hide Tool - [icon]/images/ggb/toolbar/mode_showhideobject.png[/icon] - Allows you to hide an object from view. This can be very helpful in longer constructions so you don't confuse yourself with all of the circles, points, and lines you've created.[br][br]Delete Tool - [icon]/images/ggb/toolbar/mode_delete.png[/icon] - Deletes an object. Be careful using this, as deleting one object might delete others than depend on it. For example, if you delete a point you used to create a line, then line will also disappear, as would anything you used the line to create.
Copy a Line Segment
The simplest construction is to copy a line segment. [br]Given line segment AB, copy it to the dotted line as shown in red.[br][br]1) Using the Compass tool[icon]https://www.geogebra.org/images/ggb/toolbar/mode_compasses.png[/icon], set the radius to the length of line segment AB by clicking point A then point B.[br]2) Create a new circle with a radius the length of segment AB centered at point C by clicking point C.[br]3) Create a new point where circle C and the dotted line intersect by using the Intersect tool[icon]https://www.geogebra.org/images/ggb/toolbar/mode_intersect.png[/icon] and clicking the dotted line and then the circle. [br]4) Create a new segment by selecting the Segment tool[icon]https://www.geogebra.org/images/ggb/toolbar/mode_segment.png[/icon], then click point C, then click the new point you created.[br]5) Verify the two segments are congruent by using the Segment Measure tool[icon]https://www.geogebra.org/images/ggb/toolbar/mode_distance.png[/icon]. Click point A then point B. Then click point C and the new point you created. These should be equal.
Copy an Angle
1) Start by using the Circle with Center through Point tool[icon]/images/ggb/toolbar/mode_circlepointradius.png[/icon] to create a circle with a center at point B, making sure it intersects both sides of angle ABC.[br]2) Use the Intersect tool[icon]/images/ggb/toolbar/mode_intersect.png[/icon] to mark the intersections of that circle with the sides of angle ABC.[br]3) Using the Compass tool[icon]/images/ggb/toolbar/mode_compasses.png[/icon], create a circle with a radius the same size as the one you previously made centered at point D. Do this by clicking point B, then one of the intersection points you created, then point D.[br]4) Use the Intersect tool[icon]https://www.geogebra.org/images/ggb/toolbar/mode_intersect.png[/icon] to create a point where this new circle intersects ray DF.[br]5) Use the Compass[icon]/images/ggb/toolbar/mode_compasses.png[/icon] tool to create a circle with a radius equal to the distance between the two intersection points from angle ABC. Do this by clicking one of those points, then the other. Then click the intersection point you created on ray DF in step 4.[br]6) Use the Intersect tool[icon]/images/ggb/toolbar/mode_intersect.png[/icon] to mark the intersection of this circle and the circle you created in step 3.[br]7) Use the Ray tool[icon]/images/ggb/toolbar/mode_ray.png[/icon] to create a ray from point D through the point you made in step 6.[br]8) Use the Angle Measure tool[icon]https://www.geogebra.org/images/ggb/toolbar/mode_angle.png[/icon] to verify that your new angle is congruent to the original angle. Do this by clicking on the intersection point on ray DF, then point D, then the last intersection point you created.
Copy a Triangle
1) Start by copying segment BC onto the dotted line at point D.[br]2) Copy angles B and C onto the endpoints of the segment you just created.[br]3) Use the Intersect tool[icon]/images/ggb/toolbar/mode_intersect.png[/icon] to mark where those two rays intersect. This point is the third vertex of the copied triangle.[br]4) To create the actual triangle, use the Polygon tool[icon]/images/ggb/toolbar/mode_polygon.png[/icon]. Click the vertices of your new triangle in order going around the triangle, then finally click the one you started with to complete the polygon (for example, if you started with point D, then point D also needs to be the last point you click).[br]5) Use the Distance tool[icon]/images/ggb/toolbar/mode_distance.png[/icon] and the Angle Measure tool[icon]/images/ggb/toolbar/mode_angle.png[/icon] to verify that all sides and angles of the new triangle are congruent to those of the original.[br]
Find the Center of a Circle Using Two Chords
1) Find the perpendicular bisector of one of the chords.[br]2) Find the perpendicular bisector of the other chord.[br]3) Mark where the perpendicular bisectors intersect.[br]4) Use the Distance tool[icon]/images/ggb/toolbar/mode_distance.png[/icon] to verify that this point is equidistant from the four points given.