Altitude

Definition

Altitude is the perpendicular distance from a vertex of a triangle to the line containing the opposite side. This measurement is crucial in understanding the properties and relationships within triangles, as it plays a key role in determining area and can also help establish congruence between different triangles through various geometric principles.

5 Must Know Facts For Your Next Test

1. Each triangle has three altitudes, one from each vertex to the opposite side.
2. The altitude can lie inside the triangle for acute triangles, be on the side for right triangles, and extend outside for obtuse triangles.
3. The length of an altitude is used in various calculations, including finding the area of a triangle.
4. In an equilateral triangle, all altitudes are equal in length and coincide with medians and angle bisectors.
5. Altitudes are not always congruent; their lengths depend on the angles and dimensions of the triangle.

Review Questions

• How does the altitude relate to the calculation of a triangle's area?
  • The altitude is essential for calculating a triangle's area because it represents the height from a vertex to the opposite base. The formula for area incorporates both the base and height, specifically Area = 1/2 * base * height. This shows that without knowing the altitude, you cannot accurately compute the area of a triangle, highlighting its importance in geometric calculations.
• Compare and contrast the altitudes of acute, right, and obtuse triangles regarding their positions and lengths.
  • In acute triangles, all altitudes fall inside the triangle. For right triangles, one altitude coincides with one side of the triangle while the others remain inside. In obtuse triangles, at least one altitude extends outside of the triangle. The lengths of these altitudes vary based on the specific angles and sides involved, which affects how we perceive their position relative to each type of triangle.
• Evaluate how understanding altitudes enhances your comprehension of other geometric concepts like congruence and similarity in triangles.
  • Understanding altitudes deepens your comprehension of congruence and similarity by providing insights into triangle relationships. For instance, if two triangles have equal corresponding altitudes from their respective bases, it implies that they may have proportional areas or similar dimensions. Furthermore, knowing how to construct altitudes can help demonstrate congruence through criteria such as SAS (Side-Angle-Side) or AAS (Angle-Angle-Side), as they show how angles and sides interact within different triangles.

Related terms

base:

In the context of triangles, the base refers to any side of the triangle that can serve as a reference line for measuring altitude.

triangle area: The area of a triangle can be calculated using the formula: Area = 1/2 * base * height, where height is the altitude.

orthocenter: The orthocenter is the point where all three altitudes of a triangle intersect, which can be inside, outside, or on the triangle depending on the triangle's type.