A web page and interactive applet show how to compute the perimeter of a trapezoid. A trapezoid is shown that can be resized by dragging its vertices. As you drag, the perimeter is continuously recalculated. Text on the page explains that the perimeter is the sum of the sides. For those who prefer it, in a formula that is given. 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 reference book project at http://www.mathopenref.com.

A web page and interactive applet show how to compute the perimeter of a triangle. A triangle is shown that can be resized by dragging its vertices. As you drag, the perimeter is continuously recalculated. Text on the page explains that the perimeter is the sum of the sides. For those who prefer it, in a formula that is given. 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 reference book project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate the concept of 'perpendicular'. The applet presents two intersecting line segments, one initially at right angles to the other. As you drag the end points of the line segments a message and the right-angle mark at the intersection tell if it is currently perpendicular. The goal is to show what a right angle 'looks like'. 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 perpendicular bisector of a line segment. The applet shows a fixed line and another moveable one that bisects it. As you drag either end of it, it is seen that only when they are perpendicular to each other is the definition valid. 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 how to determine of one line is perpendicular to another in coordinate geometry. The principle used is that if two lines a re perpendicular to each other the slope of one is the negative reciprocal of the other. The applet shows to lines that the user can move. The slopes are continuously calculated as you drag them, and if the they are parallel they change color. The calculation is shown on screen updated continuously as you drag. The grid, axis pointers and coordinates can be turned on and off. The calculation display 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 perpendicularity, 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 task can be implemented in a variety of ways. For a class with previous exposure to properties of perpendicular bisectors, part (a) could be a quick exercise in geometric constructions, and an application of the result. Alternatively, this could be part of an introduction to perpendicular bisectors, culminating in a full proof that the three perpendicular bisectors are concurrent at the circumcenter of the triangle, an essentially complete proof of which is found in the solution below.

This task is part of a series presenting important foundational geometric results and constructions which are fundamental for more elaborate arguments. They are presented without a real world context so as to see the important hypotheses and logical steps involved as clearly as possible.

An interactive applet and associated web page that demonstrate polygons. The applet shows a polygon which is initially an irregular convex pentagon. The user can drag any vertex and a message shows if it becomes concave. The user can also alter the number of sides from 3 to 99, the title changing to reflect it's name up to 12 sides. You can also make it regular, dragging a vertex then changes all vertices to maintain it as regular. The web page has many definitions and links to the various forms and orders of polygon. 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 that the exterior angles of a polygon always add to a constant (360 or 720 depending on your point of view). The applet shows a polygon where the user can drag the vertices to reshape it, alter the number of sides, make it convex or concave, regular or irregular. The applet shows the angle summation in real time, demonstrating that it is always constant. It also demonstrates how exterior angles are accounted for in concave polygons, where the exterior angle seems to be inside the figure. These are taken as negative angles and the sum demonstrably remains the same. 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 that the interior angles of a polygon always add to a number related only to the number of sides. The applet shows a polygon where the user can drag the vertices to reshape it, alter the number of sides, make it convex or concave, regular or irregular. The applet shows the angle summation in real time, demonstrating that only the number of sides affects the total. The text on the web page gives the formula for the total of the interior angles. 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 diagonals of a polygon. The applet shows a polygon with the diagonals drawn. The user can drag any vertex, make it regular or irregular, and change the number of sides. The applet continually computes and displays the number of diagonals. The text on the web page has the relevant formulae. 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.

"Polypad is a collection of interactive digital manipulatives for teachers and students. The unique tools and features on Polypad enable exploration, creativity, and problem solving. Students can quickly visualize their ideas and experiment with mathematical objects, while teachers can author and share complex and highly customized activities – the possibilities are endless."

Copy and paste this url to get started and find tutorials - https://polypad.amplify.com/help

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This module revisits trigonometry that was introduced in Geometry and Algebra II, uniting and further expanding the ideas of right triangle trigonometry and the unit circle.  New tools are introduced for solving geometric and modeling problems through the power of trigonometry.  Students explore sine, cosine, and tangent functions and their periodicity, derive formulas for triangles that are not right, and study the graphs of trigonometric functions and their inverses.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

Este módulo revisa la trigonometría que se introdujo en la geometría y el álgebra II, uniendo y ampliando aún más las ideas de la trigonometría del triángulo recto y el círculo unitario. Se introducen nuevas herramientas para resolver problemas geométricos y de modelado a través del poder de la trigonometría. Los estudiantes exploran funciones sinuso, coseno y tangentes y su periodicidad, derivan fórmulas para triángulos que no son correctos y estudian los gráficos de las funciones trigonométricas y sus inversos.

English Description:
This module revisits trigonometry that was introduced in Geometry and Algebra II, uniting and further expanding the ideas of right triangle trigonometry and the unit circle.  New tools are introduced for solving geometric and modeling problems through the power of trigonometry.  Students explore sine, cosine, and tangent functions and their periodicity, derive formulas for triangles that are not right, and study the graphs of trigonometric functions and their inverses.

A page that allows you to print rectangular (cartesian) graph paper. You ca control if grid lines are printed and the position of the origin. By dargging the origin into any corner a single quadrant can be printed. 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 illustrate Pythagoras' Theorem. The applet shows a right triangle that can be resized by dragging any vertex. As it is being dragged the formula is continuously recalculated to show that the the theorem always holds for any right triangle. The formula can be turned off to facilitate classroom discussions. 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 that allows the user to graphically explore the properties of a quadratic equation. Specifically, it is designed to foster an intuitive understanding of the effects of changing the three coefficients in the function. The applet shows a large graph of a quadratic (ax^2 + bx +c) and has three slider controls, one each for the coefficients a,b and c. As the sliders are moved, the graph is redrawn in real time illustrating the effects of these variations. The roots of the equation are shown both graphically and numerically, including the case where the roots are imaginary. 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 the properties of a quadrilateral. The applet shows a quadrilateral with draggable vertices. The web page has an extensive list of the various types of quadrilateral and links to other pages and applets illustrating each type. 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 define and describe the radius of a regular polygon. The applet has a polygon where the user can change the umber of sides and alter the radius by dragging a vertex. The radius is shown with it's length changing accordingly. The web page has formulae for the length of the radius given the number of sides and the apothem or side length. 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 radius of a circle. The applet shows a circle with a radius line. The radius endpoints are draggable and the figure changes to ensure the line is always a radius of the circle. Demonstrates the radius is always a constant length for a given circle. See also the entries for circumference and diameter. 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.