This line segment will be dilated with the point of dilation at ( 4 , 1 ) and a scale factor of 2 . What is the 𝑥 -coordinate of one of the endpoints of the dilated segment? Enter your answer in the box.

Answer 1

To find the \( x \)-coordinate of one of the endpoints of the dilated segment, we first need to know the original coordinates of the endpoints of the segment. However, since those coordinates aren’t provided, I’ll explain the general steps to perform the dilation.

Let’s assume one endpoint of the segment is \((x_1, y_1)\). The point of dilation is \((4, 1)\) and the scale factor is \(k = 2\).

The formula for dilation of a point \((x, y)\) with respect to a point of dilation \((x_0, y_0)\) and a scale factor \(k\) is given by:

\[ (x', y') = \left(x_0 + k(x - x_0), y_0 + k(y - y_0)\right) \]

For the point \((x_1, y_1)\):

\[ x' = 4 + 2(x_1 - 4) \] \[ y' = 1 + 2(y_1 - 1) \]

Breaking down the \(x\)-coordinate calculation:

\[ x' = 4 + 2x_1 - 8 = 2x_1 - 4 \]

Thus, the new \(x\)-coordinate of the dilated endpoint will be \(2x_1 - 4\).

If you provide the specific \(x\)-coordinate of one of the original endpoints, I can calculate the \(x\)-coordinate of the dilated endpoint for you.

Answer 2

is it right?

Answer 3

Yes, the method I provided for calculating the dilation of a point is correct. The resulting formula for the dilated \(x\)-coordinate of a point \((x_1, y_1)\) with the point of dilation at \((4, 1)\) and a scale factor of \(2\) is:

\[ x' = 2x_1 - 4 \]

However, to give a specific answer, I need the original \(x\)-coordinate of one of the endpoints.

If you provide me with the original \(x\)-coordinate \(x_1\), I can substitute that value into the formula to give you the corresponding \(x'\) for the dilated endpoint.