Triangle Classification: When Height Meets Median at Point D
Question
Look at the triangle below.
AD is the height and median.
What type of triangle is it?
Video Solution
Solution Steps
00:00Determine which type of triangle is given
00:03AD is the height of the triangle according to the given data
00:11AD is a median in the triangle, the median bisects the side
00:17A triangle where the altitude is also a median must be an isosceles triangle
00:20This is the solution
Step-by-Step Solution
We are given a triangle ABC with a vertex A, and it is noted that AD is both the height from A to the base BC and the median of the triangle. Let's analyze the situation:
Step 1: Since AD is the height, it is perpendicular to the base BC. This tells us that the line AD forms a right angle with BC.
Step 2: Being a median indicates that AD also bisects BC, meaning B and C are equidistant from D. Therefore, BD=DC.
Step 3: The dual role of AD — both a height and a median — is critical. In triangles, if a line segment from a vertex also serves as both a height and a median, the triangle is isosceles. This is due to the symmetry introduced by these properties.
Step 4: Correctly, this means two sides of triangle ABC must be equal. Specifically, since AD meets BC at its midpoint and is also perpendicular, this symmetry means side AB is equal to side AC.
Thus, triangle ABC must be isosceles. This perfectly fits the conventional definition where two sides of the triangle are equal and aligns with the properties of a line being both a height and a median.
Therefore, the solution to the problem is Isosceles.