Final answer:
To find the value of x in the given triangle ABC with centroid D, we can use the ratio of lengths of the segments of the medians. Setting up an equation using the given lengths of BD and DF will allow us to solve for x =2.
Step-by-step explanation:
Point D is the centroid of triangle ABC. The centroid of a triangle is the point where all three medians intersect.
In triangle ABC, medians AD, BE, and CF intersect at point D.
Since the point D is the centroid, it divides each median into two segments, where the ratio of the lengths of the segments is 2:1.
We are given that BD = 5x + 2 and DF = 3x.
Using the ratio of 2:1, we can set up an equation:
(5x + 2) / (3x) = 2 / 1
Solving this equation will give us the value of x.
x=2