An interactive applet and associated web page that demonstrate congruent line segments (segments that are the same length). The applet shows three line segments that are the same length. They all have draggable endpoints. As you drag any endpoint the other lines change to remain congruent with the one you are changing. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate the concept of similar triangles. Applets show that triangles are similar if the are the same shape and possibly rotated, or reflected. In each case the user can drag one triangle and see how another triangle changes to remain similar to it. The web page describes all this and has links to other related pages. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate the concept of similar polygons. Applets show that polygons are similar if the are the same shape and possibly rotated, or reflected. In each case the user can drag one polygons and see how another polygons changes to remain similar to it. The web page describes all this and has links to other related pages. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate the slope (m) of a line. The applet has two points that define a line. As the user drags either point it continuously recalculates the slope. The rise and run are drawn to show the two elements used in the calculation. The grid, axis pointers and coordinates can be turned on and off. The slope calculation can be turned off to permit class exercises and then turned back on the verify the answers. The applet can be printed as it appears on the screen to make handouts. The web page has a full description of the concept of slope, a worked example and has links to other pages relating to coordinate geometry. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that show the definition and properties of a square when applied in coordinate geometry. The applet has a square, and the user can drag any vertex to resize it. It shows how to calculate the side lengths and diagonal length given the vertex coordinates. The grid and coordinates can be turned on and off. The applet can be printed as it appears on the screen to make handouts. The web page has a full definition of a square when the coordinates of the points defining it are known, and has links to other pages relating to coordinate geometry. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate straight angles (those equal to 180 deg). The applet presents an angle (initially acute) that the user can adjust by dragging the end points of the line segments forming the angle. As it changes it shows the angle measure and a message that indicate which type of angle it is. There a software 'detents' that make it easy capture exact angles such as 90 degrees and 180 degrees The message and angle measures can be turned off to facilitate classroom discussion. The text on the page has links to other pages defining each angle type in depth. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate supplementary angles (two angles that add to 180 degrees.) The applet shows two angles which, while not adjacent, are drawn to strongly suggest visually that they add to a straight angle. Any point defining the angle scan be dragged, and as you do so, the other angle changes to remain supplementary to the one you change. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate a tangent to a circle. (not trig). The applet shows a circle and a tangent line. The center point and the tangent contact point are both draggable. As you drag each, the figure changes to ensure that the line is always tangential to the circle. The line from the center to the tangent point is shown and the angle is shown to be always 90 degrees no matter what you do. The perpendicular and its angle can be turned off for class discussion. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate the tangent to an ellipse. An ellipse is shown and a tangent line that touches it at the one point. The user can drag the tangent point around the ellipse and the tangent follows. The ellipse can be reshaped by dragging the foci. The applet shows the fact that the two lines from the foci to the tangent meet it at equal angles. The web page has the written properties of the tangent and links to other pages defining the various properties of an ellipse and to some ellipse constructions. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

A web page and interactive applet illustrating the properties of a tetragon (4 sided polygon). The applet shows a tetragon where the user can drag any vertex to reshape the polygon. User can see that the interior and exterior angles are constant in a regular tetragon, but vary in an irregular version. Controls allow the display or hiding of the diagonals, and triangles within the tetragon. The web page lists the properties of a tetragon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles. Links to pages with generalized properties of all polygons. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

A web page and interactive applet illustrating Thales Theorem (the diameter of a circle always subtends a right angle to any point on the circumference). The applet shows a circle where the user can rotate the diameter and a move a point on the circumference. The applet continuously shows the resulting right triangle, demonstrating that the theorem holds no matter how the points are moved around. Text on the page defines the theorem and relates it to other geometric entities. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate the transversal, a line that crosses two other (usually parallel) lines. The applet shows two lines that are initially parallel, and a transversal line. All eight angles thus formed are shown. As you drag the transversal end points, you can see that the angles form congruent, supplementary and complementary sets. If you move one of the parallel lines to make it non-parallel, you can see that these angles lose some of their relationship with each other. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

A web page and interactive applet showing the definition and properties of a trapezoid. The applet shows a trapezoid where the user can drag any vertex. The other points then move in such a way that the figure remains a trapezoid at all times. A control to hide the details allows a classroom discussion where students can try to infer what the properties are as it is reshaped by the discussion leader. Text on the page has the formal definition and properties of the trapezoid with links to related pages. A companion page is http://www.mathopenref.com/trapezoidarea.html showing the ways to calculate the area of a trapezoid Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that show the definition and properties of a trapezoid in coordinate geometry. The applet has a trapezoid with draggable vertices. As the user re-sizes the trapezoid the applet continuously recalculates its altitude and median from the vertex coordinates. The trapezoid can be rotated on the plane to show the more complex cases. The grid, coordinates and calculations can be turned on and off for class problem solving. The applet can be printed in the state it appears on the screen to make handouts. The web page has a full definition of a trapezoid when the coordinates of the points defining it are known, and has links to other pages relating to coordinate geometry. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page showing how to find the area and perimeter of a trapezoid from the coordinates of its vertices. The trapezoid can be either parallel to the axes or rotated. The grid and coordinates can be turned on and off. The area and perimeter calculation can be turned off to permit class exercises and then turned back on the verify the answers. The applet can be printed as it appears on the screen to make handouts. The web page has a full description of the method for determining area and perimeter, a worked example and has links to other pages relating to coordinate geometry. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate the median line of a trapezoid (line linking midpoints of non-parallel sides). The applet shows a trapezoid with all vertices draggable. As you drag any vertex, the figure changes to remain trapezoid, and the median line id continually repositioned to remain correct. It can be seen visually that the median remains parallel to the bases and a formula shows that its length is the average of the bases. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that introduce the concept of a triangle. The applet shows a triangle where the user can drag the vertices to reshape it. As it is being dragged a base and altitude are shown continuously changing. Demonstrates that the altitude may require the base to be extended. The text on the page lists the properties of a triangle and lists the various triangle types, with links to a definition of each. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

The applet shows a triangle which the user can resize by dragging any of its vertices. It shows the three perpendicular bisectors of the sides and the point where they intersect - the circumcenter. These track the changes in the triangle in real time. It shows that the circumcenter may lie outside the triangle. The associated web page describes the properties of the circumcenter and points out that it the center of the triangle's circumcircle. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

A web page and interactive applet demonstrating the concept of the incenter of a triangle. The applet shows a triangle where the user can drag the vertices around. The incenter and the angle bisector lines that generate it are continuously moved as the user drags the vertices. By experimentation the user can observe that the incenter always remains inside the triangle. The page text explains the definition of the incenter and how to find it. It describes how the incenter is also the center of the incircle of the triangle with a link to that page http://www.mathopenref.com/triangleincenter.html Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

A web page and interactive applet demonstrating the concept of the centroid of a triangle. The applet shows a triangle where the user can drag the vertices around. The centroid and the median lines that generate it are continuously moved as the user drags the vertices. By experimentation the user can observe that the centroid always remains inside the triangle. The page text explains the definition of the centroid and how to find it. It also shows that the centroid is the center of gravity of the triangle. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.