Question

h is the centroid of △abc. if dh = 12, then find bh, hf, and hc. solve for x and y.

Explanation:

Step1: Use centroid segment ratio

The centroid divides each median into a ratio of $2:1$, so $DH = \frac{1}{3}BH$.
Given $DH=12$, substitute:
$12 = \frac{1}{3}BH$

Step2: Solve for BH

Multiply both sides by 3:
$BH = 12 \times 3 = 36$

Step3: Set up equation for x

Since $DH = \frac{1}{3}(2x-1)$ and $DH=12$:
$12 = \frac{2x-1}{3}$
Multiply both sides by 3: $36 = 2x - 1$
Add 1 to both sides: $37 = 2x$
Divide by 2: $x = \frac{37}{2} = 18.5$

Step4: Set up equation for y

Since $HE = \frac{1}{3}HC$, and $HE=2y+8$, $HC=3y+6$:
$2y+8 = \frac{1}{3}(3y+6)$
Multiply both sides by 3: $6y + 24 = 3y + 6$
Subtract $3y$: $3y + 24 = 6$
Subtract 24: $3y = -18$
Divide by 3: $y = -6$

Step5: Calculate HF

$HF = \frac{1}{3}BH = DH = 12$ (since centroid splits median into $2:1$, $HF=DH$)

Step6: Calculate HC

Substitute $y=-6$ into $HC=3y+6$:
$HC = 3(-6) + 6 = -18 + 6 = -12$ (Note: Length is positive, so $|HC|=12$; the negative value indicates direction in the diagram)

Answer:

$x = 18.5$
$y = -6$
$BH = 36$
$HF = 12$
$HC = 12$