Question

lesson 5 • session 2
performing sequences of transformations
involving dilations continued
3 rotate △xyz 90° clockwise around the origin to form △xyz. then dilate the image by a scale factor of 1/2 with the center of dilation at the origin to form △xyz.
4 alana solved this problem: reflect △abc across the x - axis to form △abc. then dilate the image by a scale factor of 2 with the center of dilation at the origin to form △abc. explain and correct the mistakes shown in her work.
alana’s work
abc abc abc
a(2, 1) a(-2, 1) a(-4, 2)
b(3, 4) b(-3, 4) b(-6, 8)
c(7, 5) c(-7, 5) c(-14, 10)

Explanation:

Step1: Recall rotation rule

The rule for a 90 - degree clockwise rotation around the origin is $(x,y)\to(y, - x)$. Let's assume the coordinates of $\triangle XYZ$ are $X(x_1,y_1)$, $Y(x_2,y_2)$ and $Z(x_3,y_3)$. After a 90 - degree clockwise rotation around the origin, the coordinates of $\triangle X'Y'Z'$ are $X'(y_1,-x_1)$, $Y'(y_2,-x_2)$ and $Z'(y_3,-x_3)$.

Step2: Recall dilation rule

The rule for dilation with a scale factor $k$ centered at the origin is $(x,y)\to(kx,ky)$. Here $k = \frac{1}{2}$. So the coordinates of $\triangle X''Y''Z''$ are $X''(\frac{1}{2}y_1,-\frac{1}{2}x_1)$, $Y''(\frac{1}{2}y_2,-\frac{1}{2}x_2)$ and $Z''(\frac{1}{2}y_3,-\frac{1}{2}x_3)$.

For problem 4:

Step1: Identify reflection error

The rule for reflecting a point $(x,y)$ across the $x$ - axis is $(x,y)\to(x, - y)$. In Alana's work, when reflecting $\triangle ABC$ across the $x$ - axis to get $\triangle A'B'C'$, she incorrectly changed the $x$ - coordinate instead of the $y$ - coordinate. For example, if $A(2,1)$, then $A'(2, - 1)$ (not $A'(-2,1)$).

Step2: Identify dilation error

After getting the correct coordinates for $\triangle A'B'C'$, to dilate by a scale factor of 2 centered at the origin, we multiply the $x$ and $y$ coordinates of $\triangle A'B'C'$ by 2. For example, if $A'(2,-1)$ (the correct reflection), then $A''(2\times2,2\times(-1))=A''(4, - 2)$.

Answer:

For problem 3: You need to first apply the 90 - degree clockwise rotation rule to the coordinates of $\triangle XYZ$ and then the dilation rule with a scale factor of $\frac{1}{2}$ centered at the origin.
For problem 4:
The mistake in Alana's work for the reflection step: When reflecting across the $x$ - axis, she changed the sign of the $x$ - coordinate instead of the $y$ - coordinate.
The correct reflections are:
$A(2,1)\to A'(2, - 1)$, $B(3,4)\to B'(3, - 4)$, $C(7,5)\to C'(7, - 5)$.
The correct dilations from the correct reflected points with a scale factor of 2 centered at the origin are:
$A'(2, - 1)\to A''(4, - 2)$, $B'(3, - 4)\to B''(6, - 8)$, $C'(7, - 5)\to C''(14, - 10)$.