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GeoGebra Lab 8

Altitudes

In this lab you will construct the altitudes (or heights) of a triangle and investigate their properties.

  1. Open your saved file named GeoGebra Perpendicular Bisector Lab. Hide the perpendicular bisectors of the triangle. You should still have the intact triangle, as well as the point Circumcenter.
  2. An altitude of a triangle is a segment that joins one of the three vertices perpendicularly to a point on the line that contains the opposite side. To construct the line that contains the altitude to AB, select the perpendicular line tool and select point C and segment AB.
  3. Do the same for the other two altitude lines to their respective sides. Press escape when finished.
  4. Notice that the equations of three lines have appeared in the Algebra View to the left. As you click on each equation, the lines in the Graphic View should turn bold to denote which line the equation is representing. If you would rather have the altitude equations in slope/intercept form, right click on the equations and choose Equation \(y = mx + b\). This will be a helpful way for you to check your answers in other problems.
  5. What do you notice about the three altitude lines? Construct the point of intersection of these lines with the Intersect tool in the Point Toolbox. (GeoGebra will label this point with the next letter in the alphabet.)
  6. Change the name of the point by right clicking on the point of intersection and selecting Object Properties from the drop-down menu. In the field entitled “name” change the single-letter name to ‘Orthocenter’ (which is the name for the intersection of the altitudes of a triangle).
  7. Save this sketch as GeoGebra Altitude Lab on your computer.

Answer the following questions in a textbox on your graphics view:

  1. Make a conjecture about the three altitudes of a triangle. What do you think is always true?
  2. Test your conjecture by dragging one or more vertices around the sketch screen. What do you observe? Does this support your conjecture?
  3. What do you observe when the triangle is obtuse?
  4. What do you observe when the triangle is right?
  5. What do you observe when the triangle is acute?
  6. Why might the circumcenter and the orthocenter behave in the same ways with regards to their position relative to the triangle?
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