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$Consider /_\DEF in the figure below. The perpendicular bisectors of its sides are bar(XW), bar(YW), and bar(ZW). They meet at a single point W. (In other words, W is the circumcenter of /_\DEF.) Suppose YW=32,DE=104, and FW=68. Find EY,DW, and ZE. Note that the figure is not drawn to scale. {:[EY=◻],[DW=◻],[ZE=◻]:}$

Consider $\triangle D E F$ in the figure below. $\newline$ The perpendicular bisectors of its sides are $\overline{X W}, \overline{Y W}$ , and $\overline{Z W}$ . They meet at a single point $W$ . $\newline$ (In other words, $W$ is the circumcenter of $\triangle D E F$ .) $\newline$ Suppose $Y W=32, D E=104$ , and $F W=68$ . $\newline$ Find $E Y, D W$ , and $Z E$ . $\newline$ Note that the figure is not drawn to scale. $\newline$ $\begin{array}{l} E Y=\square \\ D W=\square \\ Z E=\square \end{array}$

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Math Problems

Grade 6

Solve one-step multiplication and division equations: word problems

Full solution

Q. Consider

\triangle D E F

in the figure below.

\newline

The perpendicular bisectors of its sides are

\overline{X W}, \overline{Y W}

, and

\overline{Z W}

. They meet at a single point

W

\newline

(In other words,

W

is the circumcenter of

\triangle D E F

\newline

Suppose

Y W=32, D E=104

, and

F W=68

\newline

Find

E Y, D W

, and

Z E

\newline

Note that the figure is not drawn to scale.

\newline

\begin{array}{l} E Y=\square \\ D W=\square \\ Z E=\square \end{array}

Circumcenter properties: Since $W$ is the circumcenter of triangle $DEF$ , it means that $W$ is equidistant from all three vertices of the triangle. This implies that $DW = FW = EW$ . We are given that $FW = 68$ , so $DW = EW = 68$ .
Finding EY: Now, we need to find $EY$ . Since $YW$ is the perpendicular bisector of $DE$ , it means that $EY = DY$ . Since $DE = 104$ , and $YW$ bisects it, $EY = DY = DE / $2$ = $104$ / $2$ = $52$.
Finding ZE: Finally, we need to find $ZE$. Since $W$ is the circumcenter, $ZW$ is the perpendicular bisector of $EF$. This means that $ZE = ZF$. Since $W$ is equidistant from the vertices, $EW = FW = 68$. Therefore, $ZE = ZF = EW = 68$.