Respuesta :
Answer:
The length of side AD is 8 units.
The length of side A'D' is 4 units
Sides CD and C'D' both have the same slope
The scale factor is 1/2
Step-by-step explanation:
we have
A(-4,0),B(-2,4),C(2,4),D(4,0)
A'(-2,0),B'(-1,2),C'(1,2),D'(2,0)
we know that
If two figures are similar, then the ratio of its corresponding sides is proportional and this ratio is called the scale factor
Verify each statement
Part 1) The length of side AD is 8 units.
The statement is true
because, the formula to calculate the distance between two points is [tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
we have
A(-4,0),D(4,0)
substitute the values
[tex]AD=\sqrt{(0-0)^{2}+(4+4)^{2}}[/tex]
[tex]AD=\sqrt{(0)^{2}+(8)^{2}}[/tex]
[tex]AD=8\ units[/tex]
Part 2) The length of side A'D' is 4 units
The statement is true
because, the formula to calculate the distance between two points is [tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
we have
A'(-2,0),D'(2,0)
substitute the values
[tex]A'D'=\sqrt{(0-0)^{2}+(2+2)^{2}}[/tex]
[tex]A'D'=\sqrt{(0)^{2}+(4)^{2}}[/tex]
[tex]A'D'=4\ units[/tex]
Part 3) The image is larger than the pre-image
The statement is false
Because
The pre-image is the trapezoid ABCD
The image is the trapezoid A'B'C'D'
Find out the scale factor
The scale factor is the ratio between corresponding sides
so
A'D'/AD=4/8=0.5
The scale factor is 0.5
therefore
The image is smaller than the pre-image
Part 4) Sides CD and C'D' both have the same slope
The statement is true
Because, the dilation does not change the shape of the figure
Verify
Find the slope CD
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
C(2,4),D(4,0)
substitute
[tex]mCD=\frac{0-4}{4-2}[/tex]
[tex]mCD=\frac{-4}{2}=-2[/tex]
Find the slope C'D'
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
C'(1,2),D'(2,0)
substitute
[tex]mC'D'=\frac{0-2}{2-1}[/tex]
[tex]mC'D'=\frac{-2}{1}=-2[/tex]
mCD=mC'D' -----> is verified
Part 5) The scale factor is 1/2
The statement is True
Because
The scale factor is the ratio between corresponding sides
so
A'D'/AD=4/8=1/2
The scale factor is 1/2
The scale factor is less than zero, so the dilation is a reduction
The true statements are:
(A) The length of side AD is 8 unit.
(B) The length of side A'D' is 4 unit.
(D) Sides CD and C'D' both have same slope.
(E) The scale factor is (1/2).
Step-by-step explanation:
Given information:
A(-4,0), B(-2,-4),C(2,4),D(4,0)
A'(-2,0),B'(-1,2),C'(1,2),D'(2,0)
As, we know:
When two figures are similar then the ratio of their sides is proportional and that ratio is called the scale factor.
(A) The length of side AD is 8 unit.
we have:
[tex]d=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]
on substituting the values in above formula:
[tex]AD=\sqrt{(0-0)^2+(4+4)^2}\\AD=\sqrt{64} \\AD=8[/tex]
Hence, the statement is true .
(B)The length of side A'D' is 4 unit.
we have:
[tex]d=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]
on substituting the values in above formula:
[tex]A'D'=\sqrt{(0-0)^2+(2+2)^2}\\A'D'=\sqrt{8} \\A'D'=4[/tex]
Hence, the statement is true .
(C)The image is larger than the pre-image.
We have
The pre-image is the trapezoid ABCD
The image is the trapezoid A'B'C'D'
Now the scale factor:
A'D'/AD=4/8
A'D'/AD=0.5
Hence , the image is smaller then the pre-image so the statement is false.
(D)Sides CD and C'D' both have same slope
Now , the slope between two point is calculated as
[tex]m=\frac{y_2-y_1}{x_2-x_1} \\[/tex]
On putting the values :
Slope of CD
[tex]m=\frac{0-4}{4-2} \\m=-2[/tex]
And similarly
Slope of C'D'
[tex]m=\frac{0-2}{2-1} \\m=-2\\[/tex]
Hence the statement is true, The slope of CD and C'D' is same.
(E)The scale factor is (1/2)
The scale factor is the ratio between the corresponding sides
So,
A'D'/AD=4/8
A'D'/AD=1/2
Hence the statement is true , The scale factor is (1/2)
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