What if you were given one or more of a triangle's angle measures? How would you determine where the triangle's altitude would be found? After completing this Concept, you'll be able to answer this type of question.
An altitude is a line segment in a triangle from a vertex and perpendicular to the opposite side, it is also known as the height of a triangle. All of the red lines are examples of altitudes:
As you can see, an altitude can be a side of a triangle or outside of the triangle. When a triangle is a right triangle, the altitude, or height, is the leg. To construct an altitude, construct a perpendicular line through a point not on the given line. Think of the vertex as the point and the given line as the opposite side.
Investigation: Constructing an Altitude for an Obtuse Triangle
Tools Needed: pencil, paper, compass, ruler
- Draw an obtuse triangle. Label it
△ABC, like the picture to the right. Extend side AC¯¯¯¯¯, beyond point A.
- Construct a perpendicular line to
AC¯¯¯¯¯, through B.
The altitude does not have to extend past side
Orthocenter: The point of concurrency for the altitudes of triangle.
Here is what the orthocenter looks like for the three triangles. It has three different locations, much like the perpendicular bisectors.
|Acute Triangle||Right Triangle||Obtuse Triangle|
|The orthocenter is inside the triangle.||The legs of the triangle are two of the altitudes. The orthocenter is the vertex of the right angle.||The orthocenter is outside the triangle.|
Which line segment is an altitude of
In a right triangle, the altitude, or the height, is the leg. If we rotate the triangle so that the right angle is in the lower left corner, we see that leg
A triangle has angles that measure
Because all of the angle measures are less than
A triangle has an angle that measures
Watch this video for help with the Examples above.
1. True or false: The altitudes of an obtuse triangle are inside the triangle.
2. Draw the altitude for the triangle shown.
3. Draw the altitude for the triangle shown.
1. Every triangle has three altitudes. For an obtuse triangle, at least one of the altitudes will be outside of the triangle, as shown in the picture at the beginning of this concept.
2. The triangle is an acute triangle, so the altitude is inside the triangle as shown below so that it is perpendicular to the base.
3. The triangle is a right triangle, so the altitude is already drawn. The altitude is
Write a two-column proof.
△ABCwith legs AB¯¯¯¯¯and AC¯¯¯¯¯ BD¯¯¯¯¯¯⊥DC¯¯¯¯¯and CE¯¯¯¯¯⊥BE¯¯¯¯¯Prove: BD¯¯¯¯¯¯≅CE¯¯¯¯¯
For the following triangles, will the altitudes be inside the triangle, outside the triangle, or at the leg of the triangle?
△JKLis an equiangular triangle.
△MNOis a triangle in which two the angles measure 30∘and 60∘.
△PQRis an isosceles triangle in which two of the angles measure 25∘.
△STUis an isosceles triangle in which two angles measures 45∘.
Given the following triangles, which line segment is the altitude?