When line AB is dilated with a scale factor of 3 and a center of dilatation at the origin, line A'B' is three times longer and three times farther from the origin than line AB. The direction and slope of the line remain unchanged.
Since line AB is dilated with a scale factor of 3 and a center of dilatation at the origin, the coordinates of A' and B' can be found by multiplying the coordinates of A and B by the scale factor of 3.
Let the coordinates of A be (x₁, y₁) and the coordinates of B be (x₂, y₂).
Then the coordinates of A' will be (3x₁, 3y₁) and the coordinates of B' will be (3x₂, 3y₂).
Since line A'B' is three times longer than line AB, the distance between A' and the origin will be three times the distance between A and the origin.
Using the distance formula, the distance between A' and the origin can be found as follows:
A'O ³= (3x₁)² + (3y₁)²
= 9(x₁² + y₁²)
The distance between A and the origin is √(x₁² + y₁²).
Therefore, the distance between A' and the origin is three times the distance between A and the origin, which can be written as 3√(x₁² + y₁²).
Similarly, the distance between B' and the origin can be found as follows:
B'O ³= (3x₂)² + (3y₂)²
= 9(x₂² + y₂²)
The distance between B and the origin is √(x₂² + y₂²).
Therefore, the distance between B' and the origin is three times the distance between B and the origin, which can be written as 3√(x₂² + y₂²).
Since line A'B' is three times farther from the origin than line AB, we can conclude that:
3√(x₂² + y₂²) = 3√(x₁² + y₁²)
This implies that x₁² + y₁² = x₂² + y₂².
The direction and slope of the line remain unchanged since dilation only affects the length and distance from the origin, but not the direction or slope of the line.
so the answer is?
The answer is that the direction and slope of the line remain unchanged. The line may be longer and farther from the origin, but its direction and slope remain the same after dilation with a scale factor of 3 and a center of dilatation at the origin.
pick two
rotation and dilation rotation and dilation reflection and translation
Alright, I will explain the concept of rotation combined with dilation and reflection combined with translation.
Rotation and Dilation:
When a shape undergoes rotation and dilation, it means that the shape is first rotated around a fixed point (called the center of rotation), and then scaled either larger or smaller with respect to that same point.
For example, let's consider a square with its center of rotation at the origin. If we rotate the square 90 degrees counterclockwise and then dilate it by a scale factor of 2, the resulting shape will be a larger square rotated at the same angle from the origin.
Reflection and Translation:
When a shape undergoes reflection and translation, it means that the shape is first reflected across a line (called the line of reflection), and then shifted horizontally or vertically by a certain distance.
For instance, let's consider a triangle reflected across the y-axis and then translated 3 units to the right. The resulting shape will have the same orientation as the original triangle, but it will be mirrored with respect to the y-axis and shifted to the right.
Please note that these combinations can have varying effects depending on the specific conditions of the transformation (e.g., center of rotation, scale factor, axis of reflection, distance of translation).