In the standard (x, y) coordinate plane, a triangle has a vertex at T(-1, 4). The triangle… — ACT

In the standard $(x, y)$ coordinate plane, a triangle has a vertex at $T(-1, 4)$ . The triangle is reflected across the $y$ -axis, and then dilated by a scale factor of $2$ with the center of dilation at the point $(3, 1)$ . What are the coordinates of the image of vertex $T$ after this sequence of transformations?

A
$(5, 7)$
B
$(2, 8)$
$(-1, 7)$ Cevap
D
$(-5, -9)$
E
$(7, -5)$

Cevap

The correct coordinates of the image of vertex

T

are

(-1, 7)

The correct coordinates are found by first reflecting the vertex across the

y

-axis to get

(1, 4)

, and then applying the dilation formula relative to the center

(3, 1)

with a scale factor of

2

, which yields

(-1, 7)

Adım Adım Çözüm

Reflect the vertex

T(-1, 4)

across the

y

-axis.

T'(1, 4)

Reflecting a point

(x, y)

across the

y

-axis maps it to

(-x, y)

, which changes the sign of the x-coordinate of

T(-1, 4)

to get

(1, 4)

Apply the dilation with a scale factor of

2

centered at

(3, 1)

to the reflected point

T'(1, 4)

T''(-1, 7)

The dilation formula for a point

(x, y)

with scale factor

k

and center

(h, k_c)

(h + k(x - h), k_c + k(y - k_c))

. Substituting

x = 1

y = 4

k = 2

h = 3

, and

k_c = 1

yields the x-coordinate

3 + 2(1 - 3) = -1

and the y-coordinate

1 + 2(4 - 1) = 7

Anahtar Kavram

Composite transformations in the coordinate plane involving reflections and non-origin-centered dilations.

Alternatif Yöntem

Instead of calculating algebraically, you can sketch the transformations on a grid. Plot

T(-1, 4)

, reflect it across the y-axis to find

T'(1, 4)

. Mark the center of dilation at

P(3, 1)

. Find the vector from

P

T'

, which is

2

units left and

3

units up. Double this vector to

4

units left and

6

units up from

P(3, 1)

, which lands at

(-1, 7)

Tahmini Süre:1m 30s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Konular

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