Two circles are similar because they have the same shape
<u>Step-by-step explanation</u>:
Circle A with center (
(
0
,
4
)
) and radius 6 has a Cartesian equation
(
x
−
0
)
2^2 + (
y
−
4
)
^2 = 6^
2
Circle B with center (
(
−
3
,
5
)
) and radius 24 has a Cartesian equation
(x+3)^2 + (y-5)^2 = 24^2
In the Cartesian plane:
translation (shift) of (
a
,
b
) vector
Every coordinate x value has a added to it; and every coordinate y value has b added to it.
Applying a translation of
→
(
0
,
4
)
to circle A gives the circle A' with the equation:
(
(
x
0
) − 0
)
^2 + (
(
y
+
4
)
−
4
)^
2 = 6^
2
→
x
^2 +
y^
2
=
6
^2
Applying a translation of →
(
3
,
−
5
) to circle B gives a circle B' with the equation:
(
(
x
+
3
)
−
3
)^
2
+
(
(
y
−
5
)
+
5
)^
2
=
24
^2
Applying a dilation of 4
to the equation of Circle A'
gives the circle A'' with equation
x
^2 + y^
2 = (
6
×
4
)
^2
→
x^
2
+
y
^2 = 24
^2
Since the equations for A'' and B' are identical
,
⇒ Circle A and Circle B are similar.
