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$Listen The coordinates of the vertices of /_\DEF are D(2,3),E(4,0), and F(1,-2). The coordinates of the vertices of /_\TVW are T(0,3), V(-2,0), and W(1,-2). Which series of transformations correctly shows that /_\DEF~=/_\TVW? a reflection over the y-axis and a translation 2 units right a reflection over the x-axis and a translation 2 units down a reflection over the x-axis and a translation 2 units up$

The coordinates of the vertices of $\triangle D E F$ are $D(2,3), E(4,0)$ , and $F(1,-2)$ . The coordinates of the vertices of $\triangle T V W$ are $T(0,3)$ , $V(-2,0)$ , and $W(1,-2)$ . $\newline$ Which series of transformations correctly shows that $\triangle D E F \cong \triangle T V W ?$ $\newline$ a reflection over the $y$ -axis and a translation $2$ units right $\newline$ a reflection over the $x$ -axis and a translation $2$ units down $\newline$ a reflection over the $x$ -axis and a translation $2$ units up

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Precalculus

Simplify radical expressions with variables

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Q. The coordinates of the vertices of

\triangle D E F

are

D(2,3), E(4,0)

, and

F(1,-2)

. The coordinates of the vertices of

\triangle T V W

are

T(0,3)

V(-2,0)

, and

W(1,-2)

\newline

Which series of transformations correctly shows that

\triangle D E F \cong \triangle T V W ?

\newline

a reflection over the

y

-axis and a translation

2

units right

\newline

a reflection over the

x

-axis and a translation

2

units down

\newline

a reflection over the

x

-axis and a translation

2

units up

Reflect over y-axis: Step $1$ : Reflect /_ $DEF$ over the y-axis. $\newline$ To reflect a point $(x, y)$ over the y-axis, change the x-coordinate to $-x$ . So: $\newline$ D $(2, 3)$ becomes D' $(-2, 3)$ , $\newline$ E $(4, 0)$ becomes E' $(-4, 0)$ , $\newline$ F $(1, -2)$ becomes F' $(-1, -2)$ .
Translate $2$ units right: Step $2$ : Translate $D'$ , $E'$ , $F'$ $2$ units to the right. $\newline$ To translate a point $(x, y)$ $2$ units right, add $2$ to the $x$ -coordinate. So: $\newline$ $D'(-2, 3)$ becomes $D''(0, 3)$ , $\newline$ $E'$ $0$ becomes $E'$ $1$ , $\newline$ $E'$ $2$ becomes $E'$ $3$ .
Compare coordinates after transformations: Step $3$ : Compare the coordinates of $\triangle TVW$ with $\triangle DEF$ after transformations. $\newline$ $T(0, 3)$ , $V(-2, 0)$ , and $W(1, -2)$ are the coordinates of $\triangle TVW$ . $\newline$ $D''(0, 3)$ , $E''(-2, 0)$ , and $F''(1, -2)$ are the coordinates of $\triangle DEF$ after transformations. $\newline$ The coordinates match, indicating that $\triangle DEF$ is congruent to $\triangle TVW$ after the specified transformations.