Question

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 aa is cj? 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 a concurrency of medians theorem identify the given int solve for cj 1. select all that apply. a. ah is a med definition of median b. f is the cent c. cj is a med simplify d cf = 30 substitute. complete the prob definition of reciprocal each statement of the solution. cf = \\( \frac { 2 } { 3 } cj \\)

Explanation:

Step1: Use the Centroid Property

The formula \(CF=\frac{2}{3}CJ\) comes from the concurrency of medians theorem (which states that the medians of a triangle intersect at a point (the centroid) two - thirds the distance from a vertex to the mid - point of the opposite side).

Step2: Substitute the value of \(CF\)

Given \(CF = 30\), substitute \(CF\) into the equation \(30=\frac{2}{3}CJ\).

Step3: Solve for \(CJ\)

Multiply both sides of the equation \(30=\frac{2}{3}CJ\) by \(\frac{3}{2}\) (the reciprocal of \(\frac{2}{3}\)). Using the property of equality (if \(a = b\), then \(a\times k=b\times k\) for \(k
eq0\)), we have \(CJ=30\times\frac{3}{2}\).

Step4: Simplify

Calculate \(30\times\frac{3}{2}=\frac{30\times3}{2}=45\).

Answer:

\(CJ = 45\)