Question

the plane used to perform dilations by multiplying coordinates is the ________. the formula used to calculate the distance between two points, ( d = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ), is called the ______. figures that have the same shape but ____ sizes are called similar figures. the proportional relationship between corresponding sides of a preimage and its image ensures ______.

Explanation:

First Question:

Dilations in coordinate geometry are performed on the coordinate plane (or Cartesian plane), where we can multiply the coordinates of points to scale the figure.

The formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) is derived from the Pythagorean theorem and is used to find the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) in a plane, and it's called the distance formula (or Pythagorean distance formula).

Similar figures are defined as figures that have the same shape but different sizes. The key property here is the difference in their sizes while the shape remains identical (corresponding angles equal, corresponding sides proportional).

Answer:

coordinate plane (or Cartesian plane)

Second Question: