The composition of transformation that maps ABCD to EHGF is "(x, y) → (x', -y') → (x' + 6, y' + 3)".
What are the transformation rules?
The basic transformation rules are:
- Translation: (x, y) → (x + a, y + b);
- Reflection: (x, y) → (x, -y) over x-axis; (x, y) → (-x, y) over y-axis;
- Dilation: (x, y) → (kx, ky)
- Rotation 90° counter-clockwise: (x, y) → (-y, x)
- Rotation 180°: (x, y) → (-x, -y)
Calculation:
The given quadrilateral ABCD has vertices,
A(-5, 2), B(-3, 4), C(-2, 4), and D(-1, 2)
The transformed quadrilateral EHGF has vertices,
E(1, 1), H(3, -1), G(4, -1), and F(5, 1)
In the map ABCD to EHGF, the transformations that took place are:
1) Reflection over x-axis
2) Translation by (x + a, y + b)
Step 1: Reflection over x-axis; (x, y) → (x, -y)
A(-5, 2) → A'(-5, (-2)) = A'(-5, -2)
B(-3, 4) → B'(-3, (-4)) = B'(-3, -4)
C(-2, 4) → C'(-2, (-4)) = C'(-2, -4)
D(-1, 2) → D'(-1, (-2)) = D'(-1, -2)
So, the reflected quadrilateral is A'B'C'D'
Step 2: Translation by (x, y) → (x + a, y + b);
The reflected quadrilateral A'B'C'D' is now translated by
A'B'C'D' → EHGF
So, (for A'(-5, -2) and E(1, 1))
x + a = 1
⇒ -5 + a = 1
⇒ a = 1 + 5
∴ a = 6
and
y + b = 1
⇒ -2 + b = 1
⇒ b = 1 + 2
∴ b = 3
Thus, the translation is (x + 6, y + 3).
Learn more about transformations here:
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