Answer:
Step-by-step explanation:
Since, old figure is the dilated form of the new figure.
For the dilation of the old figure to new by a scale factor 's',
s = [tex]\frac{\text{Side length of the new figure}}{\text{Side length of the old figure}}[/tex] = [tex]\frac{2}{4}=\frac{1}{2}[/tex]
Rule for the dilation of a point by a scale factor 'k',
(x, y) → k(x, y)
→ (kx, ky)
If the vertices of the old triangle are dilated by a scale factor of [tex]\frac{1}{2}[/tex],
Coordinates of the dilated triangle of the old triangle will be,
A(3, 4) → A'([tex]\frac{3}{2}[/tex], 2)
B(3, 8) → B'([tex]\frac{3}{2}[/tex], 4)
C(5, 4) → C'([tex]\frac{5}{2}[/tex], 2)
Now these points are rotated by an angle of 90° counterclockwise about the origin,
Rule for the rotation is,
(x, y) → (-y, x)
[tex]A'(\frac{3}{2},2)[/tex] → A"(-2, [tex]\frac{3}{2}[/tex])
B'([tex]\frac{3}{2}[/tex], 4) → B"(-4,
C'([tex]\frac{5}{2}[/tex], 2) → C"(-2,
Further these points have been reflected across x-axis,
A"(-2, [tex]\frac{3}{2}[/tex]) → P(-2, -[tex]\frac{3}{2}[/tex])
B"(-4, [tex]\frac{3}{2}[/tex]) → Q(-4, -[tex]\frac{3}{2}[/tex])
C"(-2, [tex]\frac{5}{2}[/tex]) → R(-2, -[tex]\frac{5}{2}[/tex])
Therefore, old triangle is dilated by a scale factor of [tex]\frac{1}{2}[/tex], followed by rotation of 90° counterclockwise about the origin and then reflected across x-axis to form the new triangle.
