Triangles as Geometric Figures

These resources offer online opportunities to explore several aspects of the triangle, from ideas of congruence to reflections to relationships among the triangle’s angles and sides. The final three resources deal with the the right triangle and the theorem that has proved so useful in measurement throughout centuries.

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Triangle Geometry: Angles http://www.utc.edu/~cpmawata/geom/geom.htm

This interactive exploration of triangles begins at the beginning—with angles and their classification. Students can practice their understanding and then move on to construct triangles and consider the sum of the angles of any triangle. Finally, they explore the special relationship among the sides of a right triangle—the Pythagorean theorem, demonstrated here through a Java slide show. MSP full record

Congruence of Triangles (Grades 6-8) http://nlvm.usu.edu/en/nav/frames_asid_165_g_3_t_3.html?open=instructions

With this virtual manipulative, students arrange sides and angles to construct congruent triangles. They drag line segments and angles to form triangles and flip the triangles as needed to show congruence. Options include constructing triangles given three sides (SSS), two sides and the included angle (SAS), and two angles and an included side (ASA). But the option that will motivate most discussion is constructing two triangles given two sides and a nonincluded angle (SSA). The question in this case is: Can you find two triangles that are not congruent? MSP full record

Two Equilateral Triangles http://www.ies.co.jp/math/java/geo/tworegtri/tworegtri.html

If your students have learned the theorems on triangle congruence, you may be looking for a challenging application. Here students are presented with an intricate figure showing two overlapping equilateral triangles. Because this resource is an applet, students can rotate the figure and easily see that two triangles in the figure are congruent. The challenge is to prove the triangles are congruent! The applet is from Manipula Math with Java. MSP full record

Platonic Solids (Grades 6-8) http://nlvm.usu.edu/en/nav/frames_asid_128_g_3_t_3.html?open=instructions

Using this virtual manipulative, students can examine the five platonic solids, each a polyhedron with identical regular polygonal faces. Of the five such solids discovered by the Greeks, three consist entirely of congruent triangles: the tetrahedron, the octahedron, and the 20-sided icosahedron. The applet allows students to rotate the solids, mark each face, edge, and vertex, and study how simple triangles make up these unique 3-dimensional figures. MSP full record

Transformations—Reflections http://nlvm.usu.edu/en/nav/frames_asid_297_g_3_t_3.html?open=activities

Here students can manipulate one of six geometric figures on one side of a line of symmetry and observe the effect on its image on the other side. A triangle may be selected and then translated and rotated. The line of symmetry can be moved as well, even rotated, giving more hands-on experience with reflection as students observe the effect on the image of the triangle. This applet is one of many from the National Library of Virtual Manipulatives (MSP full record), a collection of activities that encourage middle school students to explore mathematical relationships. Each activity offers how-to’s as well as instructional ideas for the teacher.

Inner Center http://www.ies.co.jp/math/java/geo/naishin/naishin.html

This applet enables students to see a fact about triangles, and then try to put the fact into words. Each triangle starts out with three small circles, one in each corner. As the student moves each circle to make it tangent to all three sides, the circle expands its radius depending on where it is moved. In this way, the student finds that no matter where the three circles start out, when they are tangent to all three sides of the triangle, they all end up in the same location. The center of that circle is called the inner center of the triangle. In this demonstration, the student sees that there is only one such center in any triangle. MSP full record

The Pythagorean Theorem http://www.arcytech.org/java/pythagoras/index.html

This site invites learners to discover for themselves “an important relationship between the three sides of a right triangle.” The site’s author, Jacobo Bulaevsky, speaks directly to students, encouraging them throughout five interactive exercises to delve deeper into the mystery. Within each exercise he gives hints that motivate and entice. The tutorial ends with information on Pythagoras of Samos and problems that rely on the theorem for their solutions. Included, also, is a page of notes for the teacher.

Pythagoras’ Theorem http://www.cut-the-knot.org/pythagoras/morey.shtml

In a quick but thorough demonstration, this applet explains each move in one proof of the theorem. The student can not only follow the steps but also back up to see an earlier step whenever necessary. Each step is a visual movement as well as an explanation in words. This page appears in Interactive Mathematics Miscellany and Puzzles. MSP full record

Three Squares: When Do Two Squares Make a New Square? http://www.figurethis.org/challenges/c45/challenge.htm

This activity opens with a diagram of two unequal squares and challenges students to find a way to construct a third square from the two. Students are encouraged to model the problem using squares of paper. The activity cleverly leads into a hands-on application of the Pythagorean theorem. The solution explains the puzzle through text and diagrams. Related questions introduce practice on the Pythagorean triples. MSP full record

For more on Pythagoras, go to Quick Take on ... Pythagoras and His Theorem.

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Copyright January 2007 — The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

This work is licensed under a Creative Commons License.