Worksheet Materials (Handout & Teacher Page)
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Materials for Offline Use

Zip for offline use: (includes handout, teachers page, 2 ggb interactivities).
Requires freeware GeoGebra and sunJava player.


Brief User interacts with construction and notices that the triangle constructed under the given conditions "appears" to be a right triangle. He then uses standard 6th grade geometric facts to "prove" that the angle is in fact 90°. Finally, in a second interactivity the user constructs the triangle.
Pre-knowledge The sum of the angles of a triangle is 180°, the sum of supplementary angles is 180° the angles at the base of an isosceles triangle are congruent (equal size).
Goal Understanding triangles, angles and logical conclusions.
Grade 6-9 (6th grade, 7th grade, pre-algebra, geometry)
Strand Measurement and Geometry, Geometry
Standards CA 6.MG.2.2,  CA Geometry 13.0, ACT PF 24-27
Keywords triangles, medians, right-triangles, construction, interactivity, geogebra  
Comments Suitable for 6th-grade on up. Can be connected to Thale's Theorem on Circles, Diameters and Right-Triangles where the "longest side" is the diameter and the median is any radius.
Source Linda Fahlberg-Stojanovska (no copyright)
Cost Activity and software is free to use
Download Zip for offline use: (includes handout, teachers page, 2 ggb interactivities)

Requires freeware GeoGebra for offline use.

Type Java Applet so requires free sunJava player
Online Activities
Theorem: Let \overline{CM}\,\, be the median to the longest side   \overline{AB}\,\, of triangle \Delta ABC .
       If \overline{CM}\,\, is exactly half the length \overline{AB}\,\, then \Delta ABC \,\,is a right-triangle.

Interactivity 1: Medians and Right Triangles   Directions for interactivity

  • Click and drag the slider a to change the length of the side \overline{AC}
  • Click and drag the slider m to change the length of the median (and the hypotenuse \overline{AB} ).
  • Click and drag point C to move and point A to rotate.
  • Click the checkboxes as desired.

Notice that angle <ACB \,\,appears to a right-angle (90°).

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Questions to think about?

  • Does triangle ABC satisfy the "hypothesis" of the theorem (no matter slider or point positions)?
  • Select checkbox: Show Angles.
  1. How many degrees in: \alpha + \beta  ?
  2. What kind of triangle is \Delta CAM  ?
  3. How many degrees in: 2 \cdot \delta + \beta ?
  4. Why does \delta =\frac{\alpha}{2} ?
  5. What kind of triangle is \Delta BMC ?
  6. Why does \frac{\alpha}{2}+ \gamma = 90^\circ ?
Finally, why does < ACB = 90^\circ ?
Interactivity 2: Do the construction!

1. Look at the construction in the window above .

2. Notice that ΔABC satisfies the theorem no matter slider or point positions. This is because of the construction!

3. Now create this construction in the window below.


  • You need to construct the solid lines!  
  • To construct \overline{CA} , click on Attach:GgbActivity/segment_len.jpg Δ, then on point C and then type in "a" (letter a - no quotes) and hit Enter.
(You can also construct \overline{CA} "by hand" with a circle with center C and radius "a" and point A - any point on this circle.)
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Page last modified on November 25, 2009, at 01:39 AM