Question

your turn
the diagram shows \\(\delta bdf\\), its medians, centroid \\(h\\), and the lengths of some of the subsegments. apply the centroid theorem to find other lengths.

1. given: \\(bh = 18\\) find \\(eh\\)
2. given: \\(bh = 18\\) find \\(be\\)
3. given: \\(ch = 4\\) find \\(cf\\)
4. given: \\(bd = 30\\) find \\(bc\\). #4 is done for you.

\\(bd = bc + cd\\) segment addition postulate
\\(30 = bc + bc\\) definition of midpoint
\\(30 = 2bc\\) simplify
\\(15 = bc\\) division property of equality

1. given: \\(bg = 11\\) find \\(bf\\)

Explanation:

Identify the properties of the centroid and medians

The problem states that \(H\) is the centroid of \(\Delta BDF\). By the Centroid Theorem, the centroid divides each median into two segments such that the distance from the vertex to the centroid is twice the distance from the centroid to the opposite side's midpoint.
For median \(BE\):

$$BH = 2 \cdot EH$$

For median \(CF\):

$$FH = 2 \cdot CH$$

For median \(DG\):

$$DH = 2 \cdot GH$$

Solve Question 1: Find \(EH\) given \(BH = 18\)

Using the Solving Linear Equations knowledge point

$$
LATEXBLOCK0
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Solve Question 2: Find \(BE\) given \(BH = 18\)

Using the Solving Linear Equations knowledge point

$$
LATEXBLOCK1
$$

Solve Question 3: Find \(CF\) given \(CH = 4\)

Using the Solving Linear Equations knowledge point

$$
LATEXBLOCK2
$$

Solve Question 5: Find \(BF\) given \(BG = 11\)

Since \(DG\) is a median, \(G\) is the midpoint of segment \(BF\).
Using the Solving Linear Equations knowledge point

$$
LATEXBLOCK3
$$

Answer:

Question 1

\(EH = 9\)

Question 2

\(BE = 27\)

Question 3

\(CF = 12\)

Question 4

\(BC = 15\)

Question 5

\(BF = 22\)