Question

a design includes a point at (5, 7, 9). the design is dilated from the origin by a scale factor of 2.5. what are the coordinates of the midpoint of the dilated segment? (2.5, 3.5, 4.5) (7.5, 9.5, 11.5) (10, 14, 18) (12.5, 17.5, 22.5)

Explanation:

Step1: Mid - point formula

The mid - point of a segment with endpoints \((x_1,y_1)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). After dilation from the origin with a scale factor \(k\), if the original point is \((x,y)\), the dilated point is \((kx,ky)\). Let the original mid - point be \((x,y)\) and after dilation with scale factor \(k = 2.5\), the new mid - point is \((2.5x,2.5y)\).

Step2: Calculate the coordinates

If the original mid - point is \((5,9)\), after dilation with a scale factor \(k = 2.5\), we use the formula for dilation \((x',y')=(k\times x,k\times y)\).
For the \(x\) - coordinate: \(x'=2.5\times5 = 12.5\)
For the \(y\) - coordinate: \(y'=2.5\times9=22.5\)

Answer:

\((12.5,22.5)\)