Given information:
- Triangle ABC with sides a = 10, b = 26, c = 32.
- CH is the altitude to the base AB.
- CM is the median to the base AB.

To find: Length of HM (HM = ?)

Solution:
1. Determine the type of triangle:
- To determine the type of triangle, we can use the triangle inequality theorem.
- According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
- In this case, we have sides a = 10, b = 26, and c = 32.
- Let's check if the triangle satisfies the theorem:

a + b > c => 10 + 26 > 32 => 36 > 32 (True)
b + c > a => 26 + 32 > 10 => 58 > 10 (True)
c + a > b => 32 + 10 > 26 => 42 > 26 (True)

- Since all three inequalities are true, the given triangle is valid.

2. Determine the length of the altitude CH:
- The altitude CH is the perpendicular distance from vertex C to the base AB.
- To find CH, we can use the formula for the area of a triangle: Area = (1/2) * base * height.
- The area of the triangle can be calculated using Heron's formula: Area = sqrt(s * (s - a) * (s - b) * (s - c)), where s is the semi-perimeter of the triangle.
- Let's calculate the area and substitute it in the formula for the altitude:

s = (a + b + c) / 2 = (10 + 26 + 32) / 2 = 68 / 2 = 34
Area = sqrt(34 * (34 - 10) * (34 - 26) * (34 - 32)) = sqrt(34 * 24 * 8 * 2) = sqrt(2^6 * 3 * 17) = 4 * sqrt(2 * 3 * 17) = 4 * sqrt(102) [approx.]
Area = (1/2) * base * height
4 * sqrt(102) = (1/2) * AB * CH
AB * CH = 8 * sqrt(102)
CH = (8 * sqrt(102)) / AB
CH = (8 * sqrt(102)) / 10
CH = 4 * sqrt(102) / 5 [approx.]

3. Determine the length of the median CM:
- The median CM is a line segment from vertex C to the midpoint of the base AB.
- In a triangle, the median divides the opposite side into two equal parts.
- Since CM is the median, it divides the base AB into two equal parts, each with length (AB/2).
- Therefore, CM = AB / 2 = 10 / 2 = 5.

4. Determine the length of HM:
- HM is a line segment from the midpoint of the base AB to the foot of the altitude CH.
- In a right-angled triangle, the altitude, median, and hypoten