Name 3-2 Additional Practice Translations What is the rule for the translation shown? 1. By CA B X D 0 2. P P S X R R Q The vertices of
are A(2, -3), B(-3, -5), and C(4, 1). For each translation, give the vertices of
. 3.
The Correct Answer and Explanation is:
To answer your question clearly and thoroughly, let’s break it down step-by-step, addressing the rule for translation and calculating the new coordinates, then providing a full explanation.
Translation Rule:
Translation means shifting every point of a figure the same distance in the same direction. The rule is typically written as:
(x, y) → (x + a, y + b)
Where:
- aa = horizontal shift (positive = right, negative = left)
- bb = vertical shift (positive = up, negative = down)
Given Triangle:
Original vertices of triangle ABC:
- A(2, -3)
- B(-3, -5)
- C(4, 1)
Let’s now determine the translation rule and find the new coordinates for three problems.
1. Translation Rule: CA → B and X → D (Incomplete Information)
This item seems to have missing information or formatting issues, so we can’t determine the exact translation rule from this without clarification. Let’s move to the next well-defined item.
2. P → P′, S → X, R → R′, Q → ?
This also seems garbled or out of context. Likely referring to a translation involving a quadrilateral PQRS, but we need clearer coordinate data.
Let’s assume sample translations for problems 3–5 using triangle ABC.
3. Translate A(2, -3), B(-3, -5), C(4, 1) by (x, y) → (x + 3, y – 2)
New coordinates:
- A′ = (2 + 3, -3 – 2) = (5, -5)
- B′ = (-3 + 3, -5 – 2) = (0, -7)
- C′ = (4 + 3, 1 – 2) = (7, -1)
4. Translate by (x, y) → (x – 2, y + 4)
New coordinates:
- A′ = (2 – 2, -3 + 4) = (0, 1)
- B′ = (-3 – 2, -5 + 4) = (-5, -1)
- C′ = (4 – 2, 1 + 4) = (2, 5)
5. Translate by (x, y) → (x + 0, y – 6)
New coordinates:
- A′ = (2, -3 – 6) = (2, -9)
- B′ = (-3, -5 – 6) = (-3, -11)
- C′ = (4, 1 – 6) = (4, -5)
Explanation:
In geometry, a translation is a type of transformation that slides every point of a figure the same distance in the same direction. It’s like moving the entire shape without rotating or flipping it. The rule for a translation is usually written as (x, y) → (x + a, y + b). The values of aa and bb determine how far and in what direction the shape moves. A positive value for aa moves the shape to the right; a negative value moves it to the left. A positive bb moves the shape up; a negative bb moves it down.
To apply a translation to a triangle, you perform the translation on each vertex of the triangle individually. For example, if triangle ABC has points A(2, -3), B(-3, -5), and C(4, 1), and we use the translation rule (x, y) → (x + 3, y – 2), then we add 3 to the x-values and subtract 2 from the y-values of each point. This gives new points A′(5, -5), B′(0, -7), and C′(7, -1).
Translations do not change the size or shape of a figure—only its position. That means the translated triangle is congruent to the original. Understanding translations is key in coordinate geometry because they help us explore symmetry, congruence, and rigid motions. They’re also used in computer graphics and real-world navigation systems.
