Altitude and Median Triangle

Altitude and median of a triangle are key geometric concepts. The altitude is a perpendicular segment from a vertex to the opposite side, while the median connects a vertex to the midpoint of the opposite side. Understanding their properties, differences, and relationships is crucial for solving various geometric problems and applications in triangle studies.

1.0What is the Median and Altitude of a Triangle?

Median of a Triangle

The median of a triangle is a line segment that extends from a vertex to the midpoint of the side opposite that vertex. Each triangle has three medians, and they intersect at a single point called the centroid, which is the center of mass of the triangle. The centroid splits each median into two segments, where the longer segment is twice as long as the shorter one.

Altitude of a Triangle

The altitude of a triangle is a perpendicular line segment extending from a vertex to the line that includes the opposite side. Each triangle has three altitudes, and they intersect at a point called the orthocenter. The length of the altitude is often used to calculate the area of the triangle.

2.0Properties of the Median of a Triangle

1. Intersection at Centroid:

• The three medians of a triangle intersect at a single point called the centroid (G).
• The centroid is also known as the center of mass or the barycenter of the triangle.

2. Centroid Divides Median:

• The centroid splits each median into two segments in a 2:1 ratio.
• The segment from the vertex to the centroid is twice as long as the segment from the centroid to the midpoint of the opposite side.

3. Equation of the Centroid:

• For a triangle with vertices A (x₁, y₁), B(x₂, y₂), and C(x₃, y₃), the coordinates of the centroid (G) are given by: $G\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)$

4. Median as a Bisector:

• Each median bisects the side to which it is drawn, dividing it into two equal parts.

5. Relation to Triangle’s Area:

• The medians split the triangle into six smaller triangles of equal area.
• This property is useful in various geometric proofs and applications.

6. Special Cases:

• In an equilateral triangle, all three medians are equal in length, and they coincide with the altitudes, perpendicular bisectors, and angle bisectors of the sides.
• In an isosceles triangle, the median drawn from the vertex to the opposite base also serves as an altitude and angle bisector.

3.0Properties of the Altitude of a Triangle

1. Intersection at Orthocenter:

• The three altitudes of a triangle intersect at a single point called the orthocenter (H).
• The orthocenter can be located inside the triangle (for an acute triangle), on the triangle (for a right triangle), or outside the triangle (for an obtuse triangle).

2. Perpendicularity:

• Each altitude of a triangle is perpendicular to the side of the triangle to which it is drawn.
• This perpendicularity is essential for defining the altitude and for calculating the area of the triangle.

3. Altitude and Area Relation:

• The length of the altitude is used to calculate the area of the triangle. The area (A) can be computed using: $A=\frac{1}{2}\times \text{ base }\times \text{ altitude }$
• Here, the base is the side to which the altitude is perpendicular.

4. Orthocenter and Euler Line:

• In an acute triangle, the orthocenter (H), circumcenter (O), and centroid (G) lie on the Euler line.
• The orthocenter is located on the Euler line, which is a fundamental line in triangle geometry.

5. Altitude Length Calculation:

• The length of the altitude from a vertex can be calculated using the formula: $h=\frac{2A}{a}$, where A is the area of the triangle, and a is the length of the base (the side opposite the vertex from which the altitude is drawn).

6. Altitude in Special Triangles:

• Right Triangle: In a right triangle, the altitude from the right-angle vertex to the hypotenuse is particularly useful for calculating the area and solving various problems.
• Equilateral Triangle: In an equilateral triangle, all 3 altitudes are equal in length and also serve as medians and angle bisectors.

7. Perpendicular Bisector Relation:

In some triangles, particularly isosceles triangles, the altitude from the vertex angle to the base also acts as the median and angle bisector.

8. Orthic Triangle:

• The triangle formed by the feet of the altitudes of a given triangle is known as the orthic triangle.
• The orthic triangle is significant in various geometric constructions and properties.

4.0Relationship Between Altitude and Median of a Triangle

1. Intersection Points:

• Altitude Intersection: The three altitudes of a triangle intersect at the orthocenter (H).
• Median Intersection: The three medians intersect at the centroid (G), which divides each median into a 2:1 ratio.

2. Special Cases in Isosceles and Equilateral Triangles:

• Isosceles Triangle: In an isosceles triangle, the median drawn from the vertex angle to the base is also an altitude and an angle bisector. It divides the base into two equal segments and is perpendicular to it.
• Equilateral Triangle: In an equilateral triangle, all 3 medians, altitudes, and angle bisectors are the same line segments. Each median is also an altitude, angle bisector, and perpendicular bisector.

5.0Difference Between Median and Altitude of Triangle

This table highlights the fundamental differences between medians and altitudes in terms of their definitions, functions, intersections, and special cases.