An interactive applet and associated web page that describe the radius of an arc and how to derive it from the width and height of the segment defined by that arc. A practical use is described for finding the radius of a circular arch given its other dimensions. 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 definition of a ray. The applet presents one point and a line that goes from that point off to infinity. As the point is dragged the line moves but it is never possible to reveal the other line end. See also the entries for line segment and line. 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 of a ray in coordinate geometry. The applet has two points that the user can drag. The two points define a ray, one of them the endpoint, the other a secondary point through which the ray passes. The grid, axis pointers 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 ray 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.

A web page and interactive applet showing the definition and properties of a rectangle. The applet shows a rectangle where the user can drag any vertex. The other points then move in such a way that the figure remains a rectangle 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 rectangle with links to related pages. A companion page is http://www.mathopenref.com/rectanglearea.html showing the ways to calculate the area of a rectangle 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 rectangle in coordinate geometry. The applet has a rectangle with draggable vertices. As the user re-sizes the rectangle the applet continuously recalculates its width, height and diagonals from the vertex coordinates. Rectangle 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 rectangle 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 rectangle from the coordinates of its vertices. The rectangle 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.

This short video and interactive assessment activity is designed to teach third graders an overview of squares and rectangles.

This task is a reasonably straight-forward application of rigid motion geometry, with emphasis on ruler and straightedge constructions, and would be suitable for assessment purposes.

The goal of this task is to give students an opportunity to experiment with reflections of triangles on a coordinate grid. Students are not prompted in the question to list the coordinates of the different triangle vertices but this is a natural extension of the task.

The goal of this task is to give students experience applying and reasoning about reflections of geometric figures using their growing understanding of the properties of rigid motions. In the case of reflecting a rectangle over a diagonal, the reflected image is still a rectangle and it shares two vertices with the original rectangle.

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles. In particular, the task has students link their intuitive notions of symmetries of a triangle with statements proving that the said triangle is unmoved by applying certain rigid transformations.

This task examines some of the properties of reflections of the plane which preserve an equilateral triangle: these were introduced in ''Reflections and Isosceles Triangles'' and ''Reflection and Equilateral Triangles I''. The task gives students a chance to see the impact of these reflections on an explicit object and to see that the reflections do not always commute.

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focussing on the class of isosceles triangles.

An interactive applet and associated web page that demonstrate reflex angles (greater than 180 deg). The applet presents an angle (initially reflex) 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 regular polygons. The applet presents a regular hexagon initially. The user can drag any vertex, which causes all the others to move in such a way as to keep the polygon regular. There are buttons to change the number of sides from 3 to 99. The apothem and circumradius are shown and continuously updated as you drag 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 relationship between the perimeter and area of a triangle. It shows that a triangle with a constant perimeter does NOT have a constant area. The applet has a triangle with one vertex draggable and a constant perimeter. As you drag the vertex, it is clear that the area varies, even though the perimeter is constant. Optionally, you can see the path traced by the dragged vertex and see that it forms an ellipse. A link takes you to a page where this effect is exploited to construct an ellipse with string and pins. The 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.

This task presents students with some creative geometric ways to represent the fraction one half. The goal is both to appeal to students' visual intuition while also providing a hands on activity to decide whether or not two areas are equal.

This task provides a context for indentifying different ways of representing half of a rectangle.

A web page and interactive applet showing the definition and properties of a rhombus. The applet shows a rhombus where the user can drag any vertex. The other points then move in such a way that the figure remains a rhombus 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 rhombus with links to related pages. A companion page is http://www.mathopenref.com/rhombusarea.html showing the ways to calculate the area of a rhombus 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 ways to calculate the area of a rhombus. The user can drag the vertices of the rhombus and the other points change automatically to ensure it remains a rhombus. A grid inside the shape allows students to estimate the area visually, then check against the actual computed area. The text on the page gives three different ways to calculate the area with a formula for each. The applet uses one of the methods to compute the area in real time, so it changes as the rhombus is reshaped with the mouse. A companion page is http://www.mathopenref.com/rhombus.html showing the definition and properties of a rhombus 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.