If the medians of a triangle are 6, 8, and 10, what is the area of the triangle?

If the medians of a triangle are 6, 8, and 10, what is the area of the triangle?

Step 1: Identify the lengths of the medians of the triangle. They are given as 6, 8, and 10.
Step 2: Calculate the semi-perimeter 's' of the triangle formed by the medians. Use the formula s = (m1 + m2 + m3) / 2, where m1, m2, and m3 are the lengths of the medians.
Step 3: Substitute the values into the formula: s = (6 + 8 + 10) / 2 = 12.
Step 4: Use the area formula for a triangle based on its medians: Area = (4/3) * √(s * (s - m1) * (s - m2) * (s - m3)).
Step 5: Substitute the values into the area formula: Area = (4/3) * √(12 * (12 - 6) * (12 - 8) * (12 - 10)).
Step 6: Calculate the values inside the square root: Area = (4/3) * √(12 * 6 * 4 * 2).
Step 7: Simplify the expression inside the square root: 12 * 6 = 72, 72 * 4 = 288, 288 * 2 = 576.
Step 8: Find the square root of 576, which is 24.
Step 9: Multiply by (4/3): Area = (4/3) * 24 = 32.
Step 10: The final area of the triangle is 32.

Medians of a Triangle – Understanding the properties and formulas related to the medians of a triangle, including how they relate to the area.
Area Calculation – Applying the formula for the area of a triangle using its medians, which is less commonly known than the standard area formulas.
Heron's Formula – Recognizing the relationship between the semi-perimeter and the sides of a triangle, and how it can be adapted for medians.