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RV-29-001 · Two transformations and an invariant area

Triangle ABC has vertices A(3, 5), B(7, 5) and C(3, 7). It is reflected in the y-axis, then translated by the vector (21)\begin{pmatrix}2\\1\end{pmatrix}.
-12-10-8-6-4-2024681012-4-224681012xyABC

(a) Find the coordinates of the final image of each vertex.

(b) Find the area of the final triangle and explain why no new measurement is needed.

Part (a) Transformations: Reflection, Translation & Invariant Area Booklet p. 75, p. 76 & p. 87
Method
1) Reflection in y-axis maps (x, y) → (−x, y)
2) Translation adds column vector (u, v)
3) Area = ½ × base × height (rigid transformations preserve area)
Worked Example (RV-29-001 · Part (a))
Find the coordinates of the final image of each vertex.
1)
Reflection changes (x, y) to (−x, y).
2)
Then add (2, 1) to each reflected point.
3)
A′(-1, 6); B′(-5, 6); C′(-1, 8)
Ans = A′(-1, 6); B′(-5, 6); C′(-1, 8)
Part (b) Transformations: Reflection, Translation & Invariant Area Booklet p. 75, p. 76 & p. 87
Method
1) Reflection in y-axis maps (x, y) → (−x, y)
2) Translation adds column vector (u, v)
3) Area = ½ × base × height (rigid transformations preserve area)
Worked Example (RV-29-001 · Part (b))
Find the area of the final triangle and explain why no new measurement is needed.
1)
Original area = ½ × 4 × 2.
2)
44 square units
3)
Both transformations preserve lengths and area.
Ans = 44 square units; reflection and translation preserve area.