Question

part 1 of 5
the medians of a triangle intersect at a point two - thirds the distance from a vertex to the midpoint of the opposite side. point f is the centroid of \\( \triangle a b c \\). if \\( c f = 30 \\), what is \\( c j \\)?
complete the problem by choosing the reason that corresponds to each statement of the solution. reasons may be used more than once or not at all.
identify the given information that will be used to solve the problem. select all that apply
\\( \square \\) a. ah is a median from the vertex a to side bc.
\\( \square \\) b. f is the centroid of \\( \triangle a b c \\).
\\( \square \\) c. cj is a median from the vertex c to side ab.
\\( \square \\) d. \\( c f = 30 \\)

Explanation:

Step1: Recall centroid property

The centroid of a triangle divides each median in a ratio of \(2:1\). That is, if \(F\) is the centroid and \(CJ\) is a median, then \(CF = \frac{2}{3}CJ\).

Step2: Solve for \(CJ\)

Given \(CF = 30\), from \(CF=\frac{2}{3}CJ\), we can solve for \(CJ\) by cross - multiplying. We get \(CJ=\frac{3}{2}CF\).
Substitute \(CF = 30\) into the formula: \(CJ=\frac{3}{2}\times30\).

$$CJ = 45$$

For the identification of given information:

• Option B: \(F\) is the centroid of \(\triangle ABC\) is used because we need the centroid property to solve the problem.
• Option D: \(CF = 30\) is the given length which we use in our calculation.

Answer:

\(CJ = 45\); Identified information: B. \(F\) is the centroid of \(\triangle ABC\), D. \(CF = 30\)