consider $\triangle tuv$ in the figure below.
the perpendicular bisectors of its sides are $\overline{ad}$, $\overline{bd}$, and $\overline{cd}$. they meet at a single point $d$.
(in other words, $d$ is the circumcenter of $\triangle tuv$.)

suppose $cd = 36$, $bv = 58$, and $ud = 60$.
find $uv$, $vd$, and $tc$.
note: the figure is not drawn to scale.

$uv = $
$vd = $
$tc = $

Question

consider $\triangle tuv$ in the figure below.
the perpendicular bisectors of its sides are $\overline{ad}$, $\overline{bd}$, and $\overline{cd}$. they meet at a single point $d$.
(in other words, $d$ is the circumcenter of $\triangle tuv$.)

suppose $cd = 36$, $bv = 58$, and $ud = 60$.
find $uv$, $vd$, and $tc$.
note: the figure is not drawn to scale.

$uv = $
$vd = $
$tc = $

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Circumcenter Properties", "Perpendicular Bisector" ], "new_concepts": [], "current_concepts": [ "Circumcenter Properties", "Perpendicular Bisector", "Pythagorean Theorem" ] } ## Find UV using perpendicular bisector properties $$ \begin{aligned} &BD \text{ is the perpendicular bisector of } UV \implies BV = UB = 58\ &UV = UB + BV = 58 + 58 = 116 \end{aligned} $$ ## Find VD using circumcenter properties $$ \begin{aligned} &D \text{ is the circumcenter of } \triangle TUV \implies TD = UD = VD\ &UD = 60 \implies VD = 60 \end{aligned} $$ ## Find TC using the Pythagorean theorem $$ \begin{aligned} &CD \text{ is the perpendicular bisector of } TU \implies \angle TCD = 90^\circ\ &\text{In right } \triangle TCD: TC^2 + CD^2 = TD^2\ &TC^2 + 36^2 = 60^2\ &TC^2 = 3600 - 1296 = 2304\ &TC = \sqrt{2304} = 48 \end{aligned} $$ Consider $\triangle TUV$ in the figure below. The perpendicular bisectors of its sides are $\overline{AD}$, $\overline{BD}$, and $\overline{CD}$. They meet at a single point $D$. (In other words, $D$ is the circumcenter of $\triangle TUV$.) Suppose $CD = 36$, $BV = 58$, and $UD = 60$. Find $UV$, $VD$, and $TC$. $UV =$ 116 $VD =$ 60 $TC =$ 48 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Circumcenter Properties" ] }

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QUESTION IMAGE

Question

Explanation:

Find UV using perpendicular bisector properties

Find VD using circumcenter properties

Find TC using the Pythagorean theorem

Answer: