directions: graph and label each figure and its image under the given dilation.
1. quadrilateral ( abcd ) with vertices ( a(-4,1) ), ( b(-2,3) ), ( c(0,-2) ), and ( d(-5,-2) ); ( k = 3 )
2. triangle ( lmn ) with vertices ( l(2,-8) ), ( m(12,8) ), and ( n(14,-4) ); ( k = 1/2 )
3. rectangle ( pqrs ) with vertices ( p(5,15) ), ( q(15,15) ), ( r(15,10) ), and ( s(5,10) ); ( k = 4/5 )
4. rhombus ( wxyz ) with vertices ( w(2,-4) ), ( x(6,-2) ), ( y(10,-4) ), and ( z(6,-6) ); ( k = 3/2 )
5. square ( mnop ) with vertices ( m(-7,-4) ), ( n(-4,-3) ), ( o(-3,-6) ), and ( p(-6,-7) ); ( k = 2 )
6. trapezoid ( defg ) with vertices ( d(-5,15) ), ( e(10,10) ), ( f(10,5) ), and ( g(-5,0) ); ( k = 1/ )

Question

Accepted Answer

### Problem 1: Quadrilateral $ABCD$ with $A(-4,1)$, $B(-2,3)$, $C(0,-2)$, $D(-5,-2)$, $k = 3$
# Explanation:
## Step 1: Dilate $A(-4,1)$
To dilate a point $(x,y)$ by a scale factor $k$, we use the formula $(kx,ky)$. For $A(-4,1)$ and $k = 3$:
$x'=3\times(-4)= - 12$, $y'=3\times1 = 3$, so $A'(-12,3)$
## Step 2: Dilate $B(-2,3)$
Using the same formula, $x'=3\times(-2)=-6$, $y'=3\times3 = 9$, so $B'(-6,9)$
## Step 3: Dilate $C(0,-2)$
$x'=3\times0 = 0$, $y'=3\times(-2)=-6$, so $C'(0,-6)$
## Step 4: Dilate $D(-5,-2)$
$x'=3\times(-5)=-15$, $y'=3\times(-2)=-6$, so $D'(-15,-6)$
# Answer:
$A'(-12, 3)$
$B'(-6, 9)$
$C'(0, -6)$
$D'(-15, -6)$

### Problem 2: Triangle $LMN$ with $L(2,-8)$, $M(12,8)$, $N(14,-4)$, $k=\frac{1}{2}$
# Explanation:
## Step 1: Dilate $L(2,-8)$
Using $(kx,ky)$ with $k=\frac{1}{2}$:
$x'=\frac{1}{2}\times2 = 1$, $y'=\frac{1}{2}\times(-8)=-4$, so $L'(1,-4)$
## Step 2: Dilate $M(12,8)$
$x'=\frac{1}{2}\times12 = 6$, $y'=\frac{1}{2}\times8 = 4$, so $M'(6,4)$
## Step 3: Dilate $N(14,-4)$
$x'=\frac{1}{2}\times14 = 7$, $y'=\frac{1}{2}\times(-4)=-2$, so $N'(7,-2)$
# Answer:
$L'(1, -4)$
$M'(6, 4)$
$N'(7, -2)$

### Problem 3: Rectangle $PQRS$ with $P(5,15)$, $Q(15,15)$, $R(15,10)$, $S(5,10)$, $k=\frac{4}{5}$
# Explanation:
## Step 1: Dilate $P(5,15)$
Using $(kx,ky)$ with $k = \frac{4}{5}$:
$x'=\frac{4}{5}\times5=4$, $y'=\frac{4}{5}\times15 = 12$, so $P'(4,12)$
## Step 2: Dilate $Q(15,15)$
$x'=\frac{4}{5}\times15 = 12$, $y'=\frac{4}{5}\times15=12$, so $Q'(12,12)$
## Step 3: Dilate $R(15,10)$
$x'=\frac{4}{5}\times15 = 12$, $y'=\frac{4}{5}\times10 = 8$, so $R'(12,8)$
## Step 4: Dilate $S(5,10)$
$x'=\frac{4}{5}\times5 = 4$, $y'=\frac{4}{5}\times10 = 8$, so $S'(4,8)$
# Answer:
$P'(4, 12)$
$Q'(12, 12)$
$R'(12, 8)$
$S'(4, 8)$

### Problem 4: Rhombus $WXYZ$ with $W(2,-4)$, $X(6,-2)$, $Y(10,-4)$, $Z(6,-6)$, $k=\frac{3}{2}$
# Explanation:
## Step 1: Dilate $W(2,-4)$
Using $(kx,ky)$ with $k=\frac{3}{2}$:
$x'=\frac{3}{2}\times2=3$, $y'=\frac{3}{2}\times(-4)=-6$, so $W'(3,-6)$
## Step 2: Dilate $X(6,-2)$
$x'=\frac{3}{2}\times6 = 9$, $y'=\frac{3}{2}\times(-2)=-3$, so $X'(9,-3)$
## Step 3: Dilate $Y(10,-4)$
$x'=\frac{3}{2}\times10 = 15$, $y'=\frac{3}{2}\times(-4)=-6$, so $Y'(15,-6)$
## Step 4: Dilate $Z(6,-6)$
$x'=\frac{3}{2}\times6 = 9$, $y'=\frac{3}{2}\times(-6)=-9$, so $Z'(9,-9)$
# Answer:
$W'(3, -6)$
$X'(9, -3)$
$Y'(15, -6)$
$Z'(9, -9)$

### Problem 5: Square $MNOP$ with $M(-7,-4)$, $N(-4,-3)$, $O(-3,-6)$, $P(-6,-7)$, $k = 2$
# Explanation:
## Step 1: Dilate $M(-7,-4)$
Using $(kx,ky)$ with $k = 2$:
$x'=2\times(-7)=-14$, $y'=2\times(-4)=-8$, so $M'(-14,-8)$
## Step 2: Dilate $N(-4,-3)$
$x'=2\times(-4)=-8$, $y'=2\times(-3)=-6$, so $N'(-8,-6)$
## Step 3: Dilate $O(-3,-6)$
$x'=2\times(-3)=-6$, $y'=2\times(-6)=-12$, so $O'(-6,-12)$
## Step 4: Dilate $P(-6,-7)$
$x'=2\times(-6)=-12$, $y'=2\times(-7)=-14$, so $P'(-12,-14)$
# Answer:
$M'(-14, -8)$
$N'(-8, -6)$
$O'(-6, -12)$
$P'(-12, -14)$

### Problem 6: Trapezoid $DEFG$ with $D(-5,15)$, $E(10,10)$, $F(10,5)$, $G(-5,0)$, $k=\frac{1}{?}$ (Assuming $k=\frac{1}{3}$ if there was a typo, but let's use $k=\frac{1}{3}$ for example)
# Explanation:
## Step 1: Dilate $D(-5,15)$
Using $(kx,ky)$ with $k=\frac{1}{3}$:
$x'=\frac{1}{3}\times(-5)=-\frac{5}{3}\approx - 1.67$, $y'=\frac{1}{3}\times15 = 5$, so $D'(-\frac{5}{3},5)$
## Step 2: Dilate $E(10,10)$
$x'=\frac{1}{3}\times10=\frac{10}{3}\approx3.33$, $y'=\frac{1}{3}\times10=\frac{10}{3}\approx3.33$, so $E'(\frac{10}{3},\frac{10}{3})$
## Step 3: Dilate $F(10,5)$
$x'=\frac{1}{3}\times10=\frac{10}{3}\approx3.33$, $y'=\frac{1}{3}\times5=\frac{5}{3}\approx1.67$, so $F'(\frac{10}{3},\frac{5}{3})$
## Step 4: Dilate $G(-5,0)$
$x'=\frac{1}{3}\times(-5)=-\frac{5}{3}\approx - 1.67$, $y'=\frac{1}{3}\times0 = 0$, so $G'(-\frac{5}{3},0)$
(Note: If the actual $k$ value is different, recalculate with the correct $k$)
# Answer:
$D'(-\frac{5}{3}, 5)$ (or decimal approximation)
$E'(\frac{10}{3}, \frac{10}{3})$ (or decimal approximation)
$F'(\frac{10}{3}, \frac{5}{3})$ (or decimal approximation)
$G'(-\frac{5}{3}, 0)$ (or decimal approximation)

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QUESTION IMAGE

Question

Explanation:

Problem 1: Quadrilateral \(ABCD\) with \(A(-4,1)\), \(B(-2,3)\), \(C(0,-2)\), \(D(-5,-2)\), \(k = 3\)

Step 1: Dilate \(A(-4,1)\)

Step 2: Dilate \(B(-2,3)\)

Step 3: Dilate \(C(0,-2)\)

Step 4: Dilate \(D(-5,-2)\)

Step 1: Dilate \(L(2,-8)\)

Step 2: Dilate \(M(12,8)\)

Step 3: Dilate \(N(14,-4)\)

Step 1: Dilate \(P(5,15)\)

Step 2: Dilate \(Q(15,15)\)

Step 3: Dilate \(R(15,10)\)

Step 4: Dilate \(S(5,10)\)

Answer:

Problem 2: Triangle \(LMN\) with \(L(2,-8)\), \(M(12,8)\), \(N(14,-4)\), \(k=\frac{1}{2}\)