Triangle Medians, Altitudes, and Inequalities

Medians

• A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.
• Every triangle has three medians.
• The medians of a triangle intersect at a point called the centroid.
• The centroid divides each median in a 2:1 ratio, with the longer segment closer to the vertex.
• The centroid is the center of mass of the triangle.

Altitudes

• An altitude of a triangle is a perpendicular line segment from a vertex to the opposite side (or an extension of the opposite side).
• Every triangle has three altitudes.
• The altitudes of a triangle intersect at a point called the orthocenter.
• The orthocenter can be inside, outside, or on the triangle, depending on the type of triangle.
• Altitudes can be used to determine the area of a triangle (area = 1/2 * base * altitude).

Inequalities in Triangles

• The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
• If a triangle has sides a, b, and c, then:
  • a + b > c
  • a + c > b
  • b + c > a
• This property is crucial in determining the possible side lengths of a triangle.

Finding Possible Side Lengths

• The triangle inequality theorem imposes restrictions on possible side lengths.
• To find possible side lengths, the sum of any two side lengths must be greater than the third side.
• Example: If two sides of a triangle have lengths 5 and 8, the third side must be greater than 8 - 5 = 3 and less than 5 + 8 = 13.
• Thus, the possible side length for the third side is between 3 and 13.
• This principle is used to determine the range of values for unknown side lengths.
• For a triangle with sides a, b, and c:
  • Find the possible range of values for each side length.
  • Based on the given information, use the inequality theorem.
  • Carefully identify and consider all valid possible combinations of side lengths.
• The restrictions imposed by the triangle inequality theorem can help determine the possible range of side lengths and thus solve problems related to finding the possible side lengths for a triangle.
• Given two sides of a triangle (e.g., a and b), the third side (c) must satisfy: | a – b | < c < a + b

Explore the fascinating concepts of medians, altitudes, and inequalities in triangles. This quiz covers the definitions, properties, and significance of these elements within triangle geometry. Test your knowledge on how these features interact and their applications in determining area and structure in triangles.