Triangle Properties: Identifying Height vs Median in Right Triangle

Video Solution

Solution Steps

00:00 Identify both the height and the median in the given triangle

00:03 The height in the right triangle is AC

00:09 BC is also a height according to the given data, forming a right angle with the line

00:15 DC is also a height according to the given data, forming a right angle with the line

00:19 DC divides AB into two equal segments, therefore it is a median

00:23 This is the solution

Step-by-Step Solution

To solve this problem, we need to analyze the roles of line segment DC in the triangle ABC:

Step 1: Identify if DC is a height
A height (altitude) is a line segment from a vertex perpendicular to the opposite side. The image suggests that line DC is vertical, with apparent perpendicularity to BC due to the presence of D near the center of the triangle, making DC a plausible height.
Step 2: Identify if DC is the median
A median divides the opposite side into two equal lengths, from a vertex. Without explicit symmetry in the image, but assuming graphic accuracy and typical diagram problems, DC can be a median if D is the midpoint.
Step 3: Confirm if DC is both a height and a median
In instances such as isosceles or equilateral triangles, a segment can be both. Given D’s position and symmetric considerations from vertex A, it's likely DC is both here.

Thus, line segment DC is both the median and the height.

Therefore, the correct answer is DC = median and height.