$\overline{dg}$, $\overline{eg}$, and $\overline{fg}$ are perpendicular bisectors. this means d, f, and e are midpoints and g is equidistant to the vertices a, b, and c

find the length of each segment.

gb =
be =
bc =
da =

Question

$\overline{dg}$, $\overline{eg}$, and $\overline{fg}$ are perpendicular bisectors. this means d, f, and e are midpoints and g is equidistant to the vertices a, b, and c

find the length of each segment.

gb =
be =
bc =
da =

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Perpendicular Bisector Theorem", "Circumcenter Properties" ], "current_concepts": [ "Perpendicular Bisector Theorem", "Circumcenter Properties", "Midpoint Properties" ] } ## Identify the circumcenter properties and find GB $$ \begin{aligned} &G \text{ is the circumcenter of } \triangle ABC \implies GA = GB = GC \ &GA = 3.9 \implies GB = 3.9 \end{aligned} $$ ## Use the Pythagorean theorem to find BE $$ \begin{aligned} &\triangle GEB \text{ is a right triangle with } \angle GEB = 90^\circ \ &BE = \sqrt{GB^2 - EG^2} \ &EG = 1.3 \text{ (since } EG = DG = 1.3 \text{ by symmetry/given diagram values)} \ &BE = \sqrt{3.9^2 - 1.3^2} = \sqrt{15.21 - 1.69} = \sqrt{13.52} \approx 3.7 \end{aligned} $$ ## Find BC and DA using midpoint properties $$ \begin{aligned} &E \text{ is the midpoint of } BC \implies BC = 2 \times BE = 2 \times 3.7 = 7.4 \ &D \text{ is the midpoint of } AB \implies DA = \frac{AB}{2} = \frac{7.4}{2} = 3.7 \end{aligned} $$ $\overline{DG}$, $\overline{EG}$, and $\overline{FG}$ are perpendicular bisectors. This means D, F, and E are midpoints and G is equidistant to the vertices A, B, and C. Find the length of each segment. GB = 3.9 BE = 3.7 BC = 7.4 DA = 3.7 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Perpendicular Bisector Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the circumcenter properties and find GB

Use the Pythagorean theorem to find BE

Find BC and DA using midpoint properties

Answer: