altitude geometry vs perpendicular bisector

Since the right bisector is perpendicular to, its slope will be the negative reciprocal of that is. The situation is pictured below, where the right bisector of is the dashed line. Find the equation of the right bisector of side. Therefore, the orthocenter is located at. It was probably unnecessary to do this, but just to be double sure that the system is consistent. Already we have, but let’s see if we can reobtain this value by solving the first two equations: So we already have the equations of the three altitudes they are:Īll that remains is to solve the above system. Īs you can observe, this is the same triangle we’ve been dealing with since the first example. Example 4įind the coordinates of the orthocenter of whose vertices are located at.

Since is perpendicular to the -axis at the point, that’s its equation. The altitude from is the blue line segment shown in the diagram below:

We can then write its equation in the form where : Thus, the slope of the altitude from will be. The altitude from will be perpendicular to. So the equation of the altitude through is. So we can write the equation of in the form where : Since the altitude is perpendicular to, the slope of is then. Ī diagram showing the altitude from ( red line segment ): In other words, if and are the slopes of two perpendicular lines, then: Since altitudes and right bisectors are perpendicular to the opposite side, finding their equations relies on an important property about slopes: if two lines are perpendicular, then the product of their slopes is NEGATIVE ONE. It may be useful to visualize the relationship between a median (RM), an altitude (RN), and a right bisector (MU) in a single diagram:Įquations of altitudes and right bisectors In the above diagram, the blue line is the right bisector of the red line segment. In particular, we can draw right bisectors for the three sides of a triangle, and they will meet at a point called the circumcenter. Such an altitude is called an external altitude, while the regular ones are called internal.Ī right bisector is a line that bisects (divides into two equal parts) a line segment and also makes an angle of with it.

Moreover, it can happen that the altitude does not meet the opposite side “directly”, unless that side is “produced” or “extended”. Unlike a median, an altitude need not go through the midpoint of the opposite side. The three altitudes meet at a point called the orthocenter. For example, if we consider shown below, the altitude from vertex is the red line segment :Īltitudes can also be drawn from vertex or vertex. In a triangle, an altitude is a perpendicular line drawn from one vertex to the opposite side.

Altitude geometry vs perpendicular bisector