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
5. given: $bg = 11$ find $bf$

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
5. given: $bg = 11$ find $bf$

Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Centroid Theorem", "Triangle Medians", "Segment Addition Postulate" ], "new_concepts": [], "current_concepts": [ "Centroid Theorem", "Triangle Medians", "Segment Addition Postulate" ] } ## Identify the medians and centroid Using the *Triangle Medians* knowledge point The diagram shows $\Delta BDF$ with medians $BE$, $CF$, and $DG$ intersecting at the centroid $H$. ## Apply the Centroid Theorem for EH Using the *Centroid Theorem* knowledge point $$ \begin{aligned} &EH = \frac{1}{2} BH \ &EH = \frac{1}{2} (18) = 9 \end{aligned} $$ ## Apply Segment Addition to find BE Using the *Segment Addition Postulate* and *Centroid Theorem* knowledge points $$ \begin{aligned} &BE = BH + EH \ &BE = 18 + 9 = 27 \end{aligned} $$ ## Apply the Centroid Theorem for CF Using the *Centroid Theorem* and *Segment Addition Postulate* knowledge points $$ \begin{aligned} &HF = 2 CH \ &HF = 2 (4) = 8 \ &CF = CH + HF = 4 + 8 = 12 \end{aligned} $$ ## Find BF given BG Using the *Triangle Medians* and *Segment Addition Postulate* knowledge points $$ \begin{aligned} &G \text{ is the midpoint of } BF \ &BF = 2 BG \ &BF = 2 (11) = 22 \end{aligned} $$ ## Question 1 $EH = 9$ ## Question 2 $BE = 27$ ## Question 3 $CF = 12$ ## Question 5 $BF = 22$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Centroid Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the medians and centroid

Apply the Centroid Theorem for EH

Apply Segment Addition to find BE

Apply the Centroid Theorem for CF

Find BF given BG

Answer:

Question 1

Question 2

Question 3

Question 5