The moment of inertia of an object measures its resistance to rotational motion. In the case of an isosceles triangle, we need to determine its moment of inertia about its base. To calculate this, we can use the parallel axis theorem.

The Parallel Axis Theorem:
The parallel axis theorem states that the moment of inertia of an object about an axis parallel to its centroidal axis is equal to the sum of the moment of inertia about its centroidal axis and the product of its mass and the square of the distance between the two axes.

Step 1: Find the centroid of the triangle:
The centroid is the point of intersection of the medians of the triangle. In an isosceles triangle, the medians and the altitude from the apex coincide. The centroid divides the median and the altitude in a 2:1 ratio. Since the base of the triangle is 150 mm, the centroid is located 100 mm from the base.

Step 2: Calculate the moment of inertia about the centroidal axis:
The moment of inertia of a triangle about its centroidal axis can be calculated using the formula:

I = (b * h^3) / 36

Where:
I = Moment of inertia about centroidal axis
b = Base of the triangle
h = Height of the triangle

In this case, the base (b) is 150 mm and the height (h) can be calculated using the Pythagorean theorem:

h = sqrt((125^2) - (75^2)) = 100 mm

Substituting these values into the formula, we get:

I = (150 * 100^3) / 36 = 416,666.67 mm^4

Step 3: Apply the parallel axis theorem:
Now, we need to find the moment of inertia about the base of the triangle using the parallel axis theorem. The distance between the centroidal axis and the base is 100 mm.

Using the parallel axis theorem, the moment of inertia about the base (I') can be calculated as:

I' = I + (m * d^2)

Where:
I' = Moment of inertia about the base
I = Moment of inertia about the centroidal axis
m = Mass of the triangle
d = Distance between the centroidal axis and the base

To find the mass of the triangle, we can consider it as a thin plate. The area of the triangle can be calculated using the formula:

Area = (1/2) * b * h

Substituting the values, we get:

Area = (1/2) * 150 * 100 = 7,500 mm^2

The mass of the triangle can be calculated by multiplying the area by the density of the material. Let's assume a density of 1 g/mm^3. Therefore, the mass (m) would be 7,500 g.

Substituting these values into the parallel axis theorem formula, we get:

I' = 416,666.67 + (7,500 * 100^2) = 916,666.67 mm^4

Therefore, the moment of inertia of the isosceles triangle about its base is 916,666.67 mm^4.