Question

1. if the scale - factor is \\(\frac{1}{2}\\), the figure will be

bigger smaller the same

1. if the scale factor is 3 the figure will be

bigger smaller the same

1. if the scale factor is 1 the figure will be

bigger smaller the same
9 - 10. with the points a(6,2), b(4, - 4), c( - 8,12) and d(2, - 8)
what are the new points if the scale factor of dilation is \\(\frac{1}{2}\\)?
a(3,1), b(, ), c( - 4,6), d(, )
ii. for each problem, create an image using the given direction. (2 points each)
11 - 12. translate the shape 6 units right
13 - 14. reflect the shape across y - axis
iii. choose the best answer. (show your solution)

1. in triangle abc, d is the mid - point of ab and e is the mid - point of bc. if ac = 3x - 15 and de = 6, what is the value of x? a. 6 b. 7 c. 9 d. 12

Explanation:

Step1: Determine size change with scale factor

If the scale factor $k < 1$, the figure is smaller. Since $k=\frac{1}{2}<1$, the figure will be smaller. If $k > 1$, the figure is bigger. Since $k = 3>1$, the figure will be bigger. If $k=1$, the figure is the same.

Step2: Calculate dilated points

For a point $(x,y)$ dilated by a scale factor $k$, the new - point is $(kx,ky)$. Given $A(6,2)$, $B(4, - 4)$, $C(-8,12)$ and $D(2,-8)$ with $k=\frac{1}{2}$.
For point $B$: $x = 4,y=-4$, new $x=\frac{1}{2}\times4 = 2$, new $y=\frac{1}{2}\times(-4)=-2$.
For point $D$: $x = 2,y=-8$, new $x=\frac{1}{2}\times2 = 1$, new $y=\frac{1}{2}\times(-8)=-4$.

Step3: Translation rule

To translate a shape 6 units to the right, we add 6 to the $x$ - coordinate of each point of the shape.

Step4: Reflection rule

To reflect a shape across the $y$ - axis, we change the sign of the $x$ - coordinate of each point of the shape.

Step5: Use mid - point theorem

In $\triangle ABC$, since $D$ is the mid - point of $AB$ and $E$ is the mid - point of $BC$, then $DE=\frac{1}{2}AC$ (Mid - point theorem). Given $AC = 3x-15$ and $DE = 6$, we have the equation $6=\frac{1}{2}(3x - 15)$.
Multiply both sides by 2: $12=3x-15$.
Add 15 to both sides: $12 + 15=3x$, so $27 = 3x$.
Divide both sides by 3: $x = 9$.

Answer:

1. smaller
2. bigger
3. the same

9 - 10. $B'(2,-2)$, $D'(1,-4)$

1. c. 9