To find the length of median CF in triangle ABC with given lengths, we can use Apollonius's theorem and properties of medians.
Given Data
- Length AB = 6
- Length AC = 8
- Length of median AD = 6
Understanding the Triangle's Configuration
- Point D is the midpoint of side BC.
- By definition of a median, AD divides triangle ABC into two smaller triangles, ABD and ACD.
Using Apollonius's Theorem
Apollonius's theorem states:
\[
AB^2 + AC^2 = 2AD^2 + 2BD^2
\]
Let BD = DC = x (since D is the midpoint):
- Then, BC = 2x.
Substituting our values into the equation:
- \(6^2 + 8^2 = 2(6^2) + 2x^2\)
Calculating the squares:
- \(36 + 64 = 72 + 2x^2\)
This simplifies to:
- \(100 = 72 + 2x^2\)
Solving for x:
- \(2x^2 = 28\)
- \(x^2 = 14\)
- \(x = \sqrt{14}\)
Calculating Length of BC
- Therefore, BC = 2x = \(2\sqrt{14}\).
Finding Length of Median CF
To find the length of median CF, we can use the formula for the length of a median, which is given by:
\[
m = \frac{1}{2} \sqrt{2AB^2 + 2AC^2 - BC^2}
\]
Substituting the known values:
- \(CF = \frac{1}{2} \sqrt{2(6^2) + 2(8^2) - (2\sqrt{14})^2}\)
Calculating:
- \(CF = \frac{1}{2} \sqrt{72 + 128 - 56}\)
- \(CF = \frac{1}{2} \sqrt{144}\)
- \(CF = \frac{1}{2} \times 12 = 6\)
Conclusion
The length of median CF is 6 units.