Question

medians and altitudes

1. match the indicated part of each triangle to the term that best describes it.
2. jacy said point v is the circumcenter of △qrs. what error did jacy likely make? what is point v actually called?
3. 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 △abc. if cf = 30, what is cj? complete the problem by choosing the reason from the box below that corresponds to each statement of the solution. reasons may be used more than once or not at all.

concurrency of medians theorem simplify.
definition of median solve for cj.
definition of reciprocal substitute.
cf = \frac{2}{3}cj
30 = \frac{2}{3}×cj
\frac{3}{2}×30 = cj
45 = cj

Explanation:

Step1: Identify the theorem for the first equation

The formula \(CF=\frac{2}{3}CJ\) is based on the Concurrency of Medians Theorem.

Step2: Substitute the value of \(CF\)

Since \(CF = 30\), substituting \(CF\) into \(CF=\frac{2}{3}CJ\) gives \(30=\frac{2}{3}\times CJ\). The reason is Substitute.

Step3: Solve for \(CJ\)

To solve \(30=\frac{2}{3}\times CJ\) for \(CJ\), we multiply both sides by \(\frac{3}{2}\) (the reciprocal of \(\frac{2}{3}\)). The reason is Solve for \(CJ\).

Step4: Simplify the expression

Calculating \(\frac{3}{2}\times30\), we get \(45 = CJ\). The reason is Simplify.

Answer:

1. \(CF=\frac{2}{3}CJ\): Concurrency of Medians Theorem
2. \(30=\frac{2}{3}\times CJ\): Substitute
3. \(\frac{3}{2}\times30 = CJ\): Solve for \(CJ\)
4. \(45 = CJ\): Simplify