Answer:
Option A , B and D are true.
The statement which are true:
The length of side AD is 4 units
The length of side A'D' is 8 units.
The scale factor is, ![\frac{1}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D)
Step-by-step explanation:
Given in figure trapezoid ABCD;
The coordinates of ABCD are:
A= (-4, 0)
B = (-2, 4)
C = (2,4)
D = (4, 0)
Since, trapezoid ABCD was dilated to create trapezoid A'B'C'D' as shown in figure;
The coordinates of A'B'C'D' are;
A' =(-2, 0)
B'=(-1, 2)
C' = (1, 2)
D' = (2, 0)
First calculate the length of AD
Using Distance formula for any two points i.e,
![\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
Since, A = (-4, 0) and D = (4, 0)
then;
Length of AD =
units
Therefore, the length of side AD is, 8 units.
Similarly find the length of A'D'.
Where A' = (-2, 0) and D' =(2,0)
Using distance formula:
Length of A'D' = ![\sqrt{(2-(-2))^2+(0-0)^2} =\sqrt{(2+2)^2}= \sqrt{4^2} = 4](https://tex.z-dn.net/?f=%5Csqrt%7B%282-%28-2%29%29%5E2%2B%280-0%29%5E2%7D%20%3D%5Csqrt%7B%282%2B2%29%5E2%7D%3D%20%5Csqrt%7B4%5E2%7D%20%3D%204)
Therefore, the length of side A'D' is, 4 units.
Now, find the slope of CD and C'D'
where C =(2, 4) , D = (4, 0) , C' = (1, 2) and D' =(2,0)
using slope formula for any two points is given by:
![Slope = \frac{y_2-y_1}{x_2-x_1}](https://tex.z-dn.net/?f=Slope%20%3D%20%5Cfrac%7By_2-y_1%7D%7Bx_2-x_1%7D)
![Slope of CD = \frac{0-4}{4-2} = \frac{-4}{2} = -2](https://tex.z-dn.net/?f=Slope%20of%20CD%20%3D%20%5Cfrac%7B0-4%7D%7B4-2%7D%20%3D%20%5Cfrac%7B-4%7D%7B2%7D%20%3D%20-2)
Similarly,
![Slope of C'D' = \frac{0-2}{2-1} = \frac{-4}{2} = -2](https://tex.z-dn.net/?f=Slope%20of%20C%27D%27%20%3D%20%5Cfrac%7B0-2%7D%7B2-1%7D%20%3D%20%5Cfrac%7B-4%7D%7B2%7D%20%3D%20-2)
Since, Sides CD and C'D' have same slope i.e, -2
Scale factor(k) states that every coordinate of the original figure has been multiplied by the scale factor.
- If k > 1, then the image is an enlargement.
- if 0<k< 1, then the image is a reduction.
- If k = 1, then the figure and the image are congruent.
The rule for dilation with scale factor(k) is;
![(x, y) \rightarrow (kx , ky)](https://tex.z-dn.net/?f=%28x%2C%20y%29%20%5Crightarrow%20%28kx%20%2C%20ky%29)
To find the scale factor:
A = (-4, 0) and A' = (-2, 0)
![(-2, 0) \rightarrow (-4k , 0)](https://tex.z-dn.net/?f=%28-2%2C%200%29%20%5Crightarrow%20%28-4k%20%2C%200%29)
On comparing we ghet;
-4k = -2
Divide -4 both sides we get;
![k = \frac{1}{2}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B1%7D%7B2%7D)
∴ The Scale factor is, ![k = \frac{1}{2}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B1%7D%7B2%7D)
Since, k < 1 which implies the image is a reduction.
Therefore, the statements which are true regarding about trapezoids are;
The length of side AD is 4 units
The length of side A'D' is 8 units.
The scale factor is, ![\frac{1}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D)