triangle XYZ has coordinates X (-2, 4), Y (-4, 7) and Z (0, 5) if the translation vector is <3,-4> what are the coordinates of the image.

Question

show your math calculations for how you find the new coordinates

Answer 1

To find the coordinates of the image of triangle XYZ after applying the translation vector \(\langle 3, -4 \rangle\), we will add the components of the translation vector to the coordinates of each point in triangle XYZ.

The given original coordinates are:

• Point X: \((-2, 4)\)
• Point Y: \((-4, 7)\)
• Point Z: \((0, 5)\)

The translation vector is \(\langle 3, -4 \rangle\), which means we will add \(3\) to the \(x\)-coordinate and \(-4\) to the \(y\)-coordinate of each point.

Let's calculate the new coordinates for each point:

1. Translate Point X (-2, 4): \[ X' = (-2 + 3, 4 - 4) = (1, 0) \]

2. Translate Point Y (-4, 7): \[ Y' = (-4 + 3, 7 - 4) = (-1, 3) \]

3. Translate Point Z (0, 5): \[ Z' = (0 + 3, 5 - 4) = (3, 1) \]

Now, we summarize the new coordinates after the translation:

• Image of Point X: \( (1, 0) \)
• Image of Point Y: \( (-1, 3) \)
• Image of Point Z: \( (3, 1) \)

Thus, the coordinates of the image after the translation are:

• \(X' (1, 0)\)
• \(Y' (-1, 3)\)
• \(Z' (3, 1)\)