Perpendicular to the base
How height is found
Altitude is central to triangle area because the area formula uses a chosen base together with its corresponding altitude. That means the height must match the selected base, not just any convenient segment.
In obtuse triangles, some altitudes fall outside the triangle because the opposite side must be extended. That makes altitude a good example of why the full geometric definition matters more than the most familiar picture.
For a triangle with base 10 units and matching altitude 6 units, the area is 30 square units. The altitude can lean across the page or land outside an obtuse triangle; it is still the segment that meets the chosen base line at a right angle.
- An altitude is a perpendicular segment from a vertex to the opposite side or its extension.
- An altitude must meet the opposite side at a right angle.
- A triangle has three altitudes, one from each vertex.
- An altitude can lie outside the triangle in an obtuse case.
A segment that measures height
Where altitudes help
- Use altitude when computing triangle area with base times height divided by two.
- Use it in orthocenter problems, since the three altitudes meet there.
- Use it in proofs where perpendicular structure from a vertex matters.
Slanted is not always height
Three altitude checks
- Do not use a slanted side as the altitude unless it is actually perpendicular to the chosen base.
- Do not forget to extend the opposite side when an obtuse triangle requires the altitude outside the figure.
- Do not treat altitude as always vertical; it is perpendicular to the chosen base, whatever the orientation.
