Question

name ______

1. be precise do you agree with the following statement? explain your reasoning.

as a result of a translation, a vertex at (3, −1) moves to (−1, 8) and a vertex at (2, 10) moves to (−4, 19).

1. why do all the coordinates of a shape change the same way with a translation?

1. a translation moves a shape’s vertex from (−2, 3) to (−8, −2). which other vertex of the same shape moved to (0, 3)?

ⓐ (−6, −2) ⓑ (6, −5) ⓒ (6, 8) ⓓ (6, −2)

Explanation:

Question 4

Step1: Find translation vector for first vertex

To find the translation vector, we calculate the difference between the new and old coordinates. For the vertex \((3, -1)\) moving to \((-1, 8)\), the horizontal change (\(\Delta x\)) is \(-1 - 3 = -4\) and the vertical change (\(\Delta y\)) is \(8 - (-1) = 9\). So the translation vector is \((-4, 9)\).

Step2: Apply translation vector to second vertex

Now, apply this translation vector to the vertex \((2, 10)\). The new \(x\)-coordinate should be \(2 + (-4) = -2\) and the new \(y\)-coordinate should be \(10 + 9 = 19\). But the statement says it moves to \((-4, 19)\), which has a different \(x\)-coordinate. So the translation is not consistent.

A translation is a rigid transformation that slides a shape without rotating or resizing it. In a translation, every point of the shape is moved the same distance and in the same direction. So the change in the \(x\)-coordinate (horizontal shift) and \(y\)-coordinate (vertical shift) is the same for all points. For example, if a shape is translated 3 units left and 2 units up, every vertex's \(x\)-coordinate decreases by 3 and \(y\)-coordinate increases by 2. This preserves the shape and size, and ensures all coordinates change the same way.

Step1: Find translation vector

First, find the translation vector from the vertex \((-2, 3)\) to \((-8, -2)\). The \(\Delta x = -8 - (-2) = -6\) and \(\Delta y = -2 - 3 = -5\). So the translation vector is \((-6, -5)\).

Step2: Reverse translation to find original vertex

We know the translated vertex is \((0, 3)\). To find the original vertex, we reverse the translation (add 6 to \(x\), add 5 to \(y\)). So the original \(x\)-coordinate is \(0 + 6 = 6\) and the original \(y\)-coordinate is \(3 + 5 = 8\)? Wait, no—wait, the translation vector is \((\Delta x, \Delta y)=(-6, -5)\), so to get from original \((x, y)\) to translated \((x', y')\), we have \(x' = x + \Delta x\) and \(y' = y + \Delta y\). So to find \(x\) when \(x' = 0\), \(x = x' - \Delta x = 0 - (-6) = 6\). To find \(y\) when \(y' = 3\), \(y = y' - \Delta y = 3 - (-5) = 8\)? Wait, no, wait the original vertex was \((-2, 3)\) and translated to \((-8, -2)\). So the translation is \(x\) goes from \(-2\) to \(-8\) (change of \(-6\)), \(y\) from \(3\) to \(-2\) (change of \(-5\)). So if a vertex is translated to \((0, 3)\), its original vertex is \(x = 0 - (-6) = 6\)? Wait no, wait: original \(x\) + \(\Delta x\) = translated \(x\). So original \(x\) = translated \(x\) - \(\Delta x\) = \(0 - (-6) = 6\). Original \(y\) = translated \(y\) - \(\Delta y\) = \(3 - (-5) = 8\)? But that's option C? Wait no, wait maybe I messed up. Wait the translation vector is \((-6, -5)\), so to go from original to translated, we subtract 6 from \(x\) and subtract 5 from \(y\). So if translated is \((0, 3)\), original is \(0 + 6 = 6\) (for \(x\)) and \(3 + 5 = 8\) (for \(y\))? But the options are A \((-6, -2)\), B \((6, -5)\), C \((6, 8)\), D \((6, -2)\). Wait, no—wait the problem says "Which other vertex of the same shape moved to \((0, 3)\)?" So we need to find the original vertex (before translation) that, after translation (using the same vector), becomes \((0, 3)\). The translation vector is \(\Delta x = -8 - (-2) = -6\), \(\Delta y = -2 - 3 = -5\). So original vertex \((x, y)\) satisfies \(x + (-6) = 0\) and \(y + (-5) = 3\). Solving for \(x\): \(x = 0 + 6 = 6\). Solving for \(y\): \(y = 3 + 5 = 8\). Wait, but that's option C? Wait no, wait the original vertex that moves to \((0, 3)\) is \((6, 8)\)? But let's check: translate \((6, 8)\) by \((-6, -5)\): \(6 - 6 = 0\), \(8 - 5 = 3\). Yes, that works. Wait but let's check the first vertex: \((-2, 3)\) translated by \((-6, -5)\) is \(-2 -6 = -8\), \(3 -5 = -2\), which matches. So the original vertex is \((6, 8)\), which is option C? Wait no, the options are A \((-6, -2)\), B \((6, -5)\), C \((6, 8)\), D \((6, -2)\). So the answer is C? Wait, no—wait maybe I made a mistake. Wait the translation is from original to translated: original \((x, y)\) → translated \((x - 6, y - 5)\). So if translated is \((0, 3)\), then \(x - 6 = 0\) ⇒ \(x = 6\); \(y - 5 = 3\) ⇒ \(y = 8\). So original vertex is \((6, 8)\), which is option C.

Answer:

No, because the translation vector from \((3, -1)\) to \((-1, 8)\) is \((-4, 9)\), and applying it to \((2, 10)\) gives \((-2, 19)\), not \((-4, 19)\).

Question 5