Investigating Geometry with GeoGebra

Henrico County Public Schools

What are the relationships between the altitudes and angles of a triangle?

READ the instructions on the right and INTERACT with the GeoGebra applet below.

Sorry, the GeoGebra Applet could not be started. Please install Java 1.4 (or later) to use this page.

Given an arbitrary ΔABC, the altitudes (a line from a vertex that is perpendicular to the opposite side) intersect (concurrency) at a single point, the orthocenter.

Investigation Steps

Click on the reset icon icon to reset the diagram.

 Step 1 Close 
  1. Create segments AG, GE, BG, and GF.

  2. Measure the lengths of these segments and determine their ratios. Record your measures.

  3. Ctrl or right click the segments, in the pop-up menu, select Properties. Click on Show Label:Value.

  4. Is this ratio the same for all three medians? Is the the conjecture correct?
 Step 2 Close 
  1. Using the Text tool, text tool click on the screen, delete the quotes, and type Area[A,B,C] to find the area of ΔABC.

  2. Find the area of ΔAGB

  3. Find the ratio (Area of ΔAGB)/(Area of ΔABC).
 Step 3 Close 
  1. Repeat to find the ratios (Area of ΔAGC/Area of ΔABC) and (Area of ΔBGC/Area of ΔABC).

  2. Move the blue points to change the triangle. Does the ratios remain the same?

  3. Write a conjecture of your findings to your instructor.