If segment EF has endpoints at E(-3,10) and F (5,6) and is dilated about the origin by a factor of 7 Find the new length.?
Answers
Answer:Scale Factor for dilation is
3
.
Useful observations involving Dilation:
Isometry refers to a linear transformation which preserves the length.
Dilation is NOT an isometry. It creates similar figures only.
Here
¯¯¯¯¯¯
A
B
is the pre-image and after dilation,
¯¯¯¯¯¯¯¯¯
A
'
B
'
is called the image.
The absolute value of the scale factor (k),
with the constraint
0
<
k
<
1
,
reduces the line segment
¯¯¯¯¯¯
A
B
,
enlarges if otherwise.
Each point on the line segment
¯¯¯¯¯¯
A
B
will get 3 times as far from the Center of Dilation,
(
4
,
3
)
since the scale factor is
3
.
Dilation preserves the angle of measure.
Note that the pre-image and the image are parallel.
Observe that the points (center of dilation
C
, A and A') collinear.
And, the points (C, B and B') are also collinear.
¯¯¯¯¯¯
A
B
∣
∣
¯¯¯¯¯¯¯¯¯
A
'
B
'
, since we have congruent corresponding angles.
Also, from
C
(
4
,
3
)
, move up 4 units on the y-axis, and 2 units left on the x-axis to reach the end-point
A
(
2
,
7
)
.
Move (4 x 3 = 12 units) up on the y-axis, and (2 x 3 = 6 units) left on the x-axis tor reach the end-point of
A
'
B
'
(
−
2
,
15
)
.
Similarly,
from
C
(
4
,
3
)
, move one unit up on the y-axis and one unit right on the x-axis, to reach point
B
(
5
,
4
)
.
From
C
(
4
,
3
)
, move (1 x 3 = 3 units) on the y-axis, (1 x 3 = 3 units) to the right on the x-axis, to reach the point
B
'
(
7
,
6
)
.
New end-points:
A
'
(
−
2
,
15
)
and
B
'
(
7
,
6
)
Find the length of
¯¯¯¯¯¯¯¯¯
A
'
B
'
, using distance formula:
D
=
√
(
x
2
−
x
1
)
2
+
(
y
2
−
y
1
)
2
⇒
D
=
√
(
7
−
(
−
2
)
2
)
+
(
6
−
15
)
2
⇒
D
=
√
9
2
+
(
−
9
)
2
⇒
D
=
√
162
⇒
D
≈
12.72792
¯¯¯¯¯¯¯¯¯
A
'
B
'
≈
12.73 units
Hope this solution is helpful.
Step-by-step explanation:
