6-2:  Altitudes and Perpendicular Bisectors

Using some of the construction methods used in previous sections and definitions covered in this section we will GSP to construct the altitudes and perpendicular bisectors of a given triangle.

Given the following triangle

Construct the altitudes and perpendicular bisectors of the triangle.

Step by Step Instruction

a.  To construct the altitudes of the given triangle we have to construct a segment perpendicular to each side through the vertex opposite of the side.  To do this we will use our circle construction tool.

First construct a circle with vertex A as the center such that it intersects line segment BC in two places.  (Select your circle construction tool, start with the point at vertex A and expand the circle till it intersects line segment BC.

Label the points where the circle intersects line segment BC, point D and point E

b. We can now hide this circle. (Highlight the circle, from the "Display" menu select "Hide Object") Then we want to construct a circle in the same manner with center as point D; this circle should have a radius more than half the distance from pt D to pt E but intersects line segment DE.

c. Next we need to construct a circle with center point E and radius equal to the circle we just constructed. To make it accurate let's construct a line segment from point D to the intersection point from the circle and line segment DE. Now that we have a radius to go off of we can construct our second circle. (Highlight our radius and point E, from the "Construct" menu select "Circle by Center + Radius")

d. Identify the point of intersection of the two circles that lies outside the triangle then construct a segment from this point to vertex A.

e. Now identify the point where our new line segment intersects side BC and construct a line segment from this point to vertex A. Hide all but the triangle and the new line segment. What we have is the given triangle with one of it's altitudes.

Let's explore this altitude. First let's prove that it is actually the altitude by proving the segment is perpendicular to side BC. We do this by measuring the angles on either side and verifying they are right angles. (Highlight points A, F and C, from the "Measure" menu select "Angle" then do the same for angle AFB.)

Now that we have shown that it is the altitude let's show that the altitude doesn't always lie in the interior of the triangle. First let's extend line segment BC and line segment AF into lines.

To show the altitude doesn't always lie in the interior of the triangle we can move vertex A around to change the shape of the triangle.

Here we have shown that if we have a right triangle the altitude is actually the side of the triangle

Now we have shown that altitudes can sometimes lie outside of the triangle

So, using the previous method construct the other altitudes of the given triangle. (Hint: Extend the sides of the triangles to lines)

Our altitudes are line segments AF, BG, and CH.
Now let's discuss the perpendicular bisectors.

a. Using previous construction methods locate the midpoint for line segment BC.

Now we have to construct a line perpendicular to side BC through the midpoint

b. We first construct a circle with center as point I that intersects side BC in two places and identify the two points of intersection.

c. Now construct a circle with center pt J that intersects side BC between points I and K. You can hide the first circle to eliminate any confusion.

d. Then construct a circle with center point K and radius equal to our last circle. Then identify the points where the two circles intersect.

e. Finally we construct a line through our two intersection points and hide our circles and excess points and we have our perpendicular bisector

Now construct the other perpendicular bisectors using the same method