Triangle ABC

Question

A
B
C
has vertices point A(2, 1)
A
(
2
,

1
)
, point B(−2, 2)
B
(
−
2
,

2
)
, and point C(−3,−1)
C
(
−
3
,
−
1
)
. After dilation, triangle A′B′C′
A
′
B
′
C
′
has vertices point A′(4, 2)
A
′
(
4
,

2
)
, point B′(−4, 4)
B
′
(
−
4
,

4
)
, and point C′(−6,−2)
C
′
(
−
6
,
−
2
)
. Determine the scale factor for the dilation of triangle ABC
A
B
C
with a center point of dilation at the origin (0, 0 )
(
0
,

0

)
.

Answer 1

To determine the scale factor for the dilation of triangle \( ABC \) to triangle \( A'B'C' \) centered at the origin (0, 0), we can compare the coordinates of corresponding vertices.

The formula for dilation centered at the origin with scale factor \( k \) is given by:

\[ (x, y) \rightarrow (kx, ky) \]

Now, let's examine the coordinates of the vertices of triangles \( ABC \) and \( A'B'C' \):

1. For point \( A \) and point \( A' \):
   • Coordinates of \( A \) are \( (2, 1) \)
   • Coordinates of \( A' \) are \( (4, 2) \)

Applying the dilation formula: \[ (k \cdot 2, k \cdot 1) = (4, 2) \] From the x-coordinates: \[ k \cdot 2 = 4 \implies k = \frac{4}{2} = 2 \] From the y-coordinates: \[ k \cdot 1 = 2 \implies k = \frac{2}{1} = 2 \]

2. For point \( B \) and point \( B' \):
   • Coordinates of \( B \) are \( (-2, 2) \)
   • Coordinates of \( B' \) are \( (-4, 4) \)

Applying the dilation formula: \[ (k \cdot -2, k \cdot 2) = (-4, 4) \] From the x-coordinates: \[ k \cdot -2 = -4 \implies k = \frac{-4}{-2} = 2 \] From the y-coordinates: \[ k \cdot 2 = 4 \implies k = \frac{4}{2} = 2 \]

3. For point \( C \) and point \( C' \):
   • Coordinates of \( C \) are \( (-3, -1) \)
   • Coordinates of \( C' \) are \( (-6, -2) \)

Applying the dilation formula: \[ (k \cdot -3, k \cdot -1) = (-6, -2) \] From the x-coordinates: \[ k \cdot -3 = -6 \implies k = \frac{-6}{-3} = 2 \] From the y-coordinates: \[ k \cdot -1 = -2 \implies k = \frac{-2}{-1} = 2 \]

In all cases, we have found that the scale factor \( k \) is 2.

Thus, the scale factor for the dilation is: \[ \boxed{2} \]