# Triangles - Medians and Right Triangles

Worksheet Materials (Handout & Teacher Page) 2-page printable   pdf  or  doc   (opens in new window)

Materials for Offline Use

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

Metadata

 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: tri_median_right.zip (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°).

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. Help 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.) Click on Attach:GgbActivity/circle_rad.jpg Δ, then on point C and type in "m". Do the same for point A. Then find the intersection point and name it M ... This browser does not have a Java Plug-in. Get the latest Java Plug-in here.

Related themes:

Page last modified on November 25, 2009, at 01:39 AM