Question

Kim says that the medians of a triangle are always the same length. Hector says that they are never the same length. Who is correct? Explain your reasoning.

Short answer

Kim is correct for equilateral triangles, and Hector is correct for scalene triangles.

Step-by-step solution

Understanding Medians of a Triangle

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. In any triangle, there are three medians.

Investigating Kim's Statement

Kim claims that the medians are always of the same length. To verify, consider an equilateral triangle, where all sides and angles are equal. In this scenario, all medians are equal in length, meaning Kim is correct only in specific cases like equilateral triangles.

Examining Hector's Statement

Hector claims that the medians are never the same length. Consider a scalene triangle, with sides and angles of different measures. In scalene triangles, the medians have different lengths. Therefore, Hector's statement holds true for scalene triangles.

Comparing the Statements

Kim's claim that medians are always the same length is true only for equilateral triangles. Hector's claim that medians are never the same length is true for scalene triangles. Since both have scenarios where they are correct, neither statement is universally true.

Key concepts

Equilateral Triangle Properties and Medians

Equilateral triangles are a special type of triangle where all three sides are of equal length. This characteristic directly influences other aspects of the triangle, including its angle measures and medians. In an equilateral triangle, each angle measures 60 degrees, making it very symmetrical.
The medians, which are the line segments that extend from each vertex to the midpoint of the opposite side, are also equal in length in an equilateral triangle. This occurs because the sides are all the same length, ensuring the midpoints are equidistant. Consequently, the medians also act as the angle bisectors, altitudes, and perpendicular bisectors in this triangle type.

• All sides equal
• All medians equal in length
• Each median passes through the centroid, the common intersection point of the medians
• Acts as angle bisectors, altitudes, and perpendicular bisectors as well

Understanding these properties helps visualize why, in an equilateral triangle, the medians are all of the same length.

Scalene Triangle Characteristics

Scalene triangles are quite different from equilateral triangles as none of their sides or angles are identical. Each side length and angle measurement varies, giving these triangles a more irregular appearance. Because of this variation, the medians also differ in length.
In a scalene triangle, each median still connects a vertex to the midpoint of the opposing side, but due to the varying lengths of the triangle's sides, the medians will not be equal. The position of the centroid will be closer or further from particular vertices, altering the medians' lengths. This distinct property of scalene triangles highlights the diversity among triangle types.

• All sides and angles are different
• Medians are of varying lengths
• The centroid divides each median into a ratio of 2:1

These characteristics explain why in a scalene triangle, unlike an equilateral triangle, the lengths of the medians are never the same.

General Triangle Properties and Medians

Triangles have various properties that apply regardless of whether they are equilateral, isosceles, or scalene. One such property is having three sides, three angles, and three vertices. The internal angles always add up to 180 degrees, providing the foundational geometry for other calculations.
Medians are significant features in triangle geometry. Each triangle has three medians, and they often intersect at a point called the centroid. The centroid has special properties, like dividing each median into two segments—one of which is twice as long as the other.

• Sum of internal angles equals 180 degrees
• Three medians intersect at the centroid
• Centroid divides medians in a 2:1 ratio

These general properties are visible in both equilateral and scalene triangles, albeit manifesting differently due to the unique characteristics and structures of these triangle types.