Triangle `A 1,3 B 2,5 C 4,3 ` is dilated by a scale factor of 2 using point `B` as the center of dilation.

Give the vertices of triangle `A'B'C'`

B is the center of dilation ... so B' equals B

Ay is 2 less than By ... A'y is 4 less than B'y

the other coordinates work the same way
... the relative distances from B are doubled

To dilate a triangle with a scale factor of 2 using point B as the center of dilation, we multiply the coordinates of each vertex by 2.

1. Start with the original coordinates of the triangle:
A(1,3), B(2,5), C(4,3).

2. To find the coordinates of A', multiply the coordinates of A by 2:
A' = (2 * 1, 2 * 3) = (2, 6).

3. To find the coordinates of B', we don't need to perform any calculations since B is the center of dilation. Therefore, B' will have the same coordinates as B:
B' = (2, 5).

4. To find the coordinates of C', multiply the coordinates of C by 2:
C' = (2 * 4, 2 * 3) = (8, 6).

Therefore, the vertices of triangle A'B'C' are:
A'(2, 6), B'(2, 5), C'(8, 6).

To find the vertices of triangle A'B'C' after dilating triangle ABC by a scale factor of 2 using point B as the center of dilation, we can follow these steps:

1. Find the distance between point A and point B (AB).
- Using the coordinates of A (1,3) and B (2,5), we can calculate the distance AB using the distance formula: √((x2 - x1)^2 + (y2 - y1)^2).
- AB = √((2 - 1)^2 + (5 - 3)^2) = √(1^2 + 2^2) = √5.

2. Multiply the distance AB by the scale factor of 2 to get the new distance A'B'.
- A'B' = AB * 2 = √5 * 2 = 2√5.

3. Determine the direction of A' relative to B.
- Since we are dilating using point B as the center, A' will be twice as far away from B as A was. Therefore, A' will be in the same direction as A from B.

4. Find the coordinates of A' using point B as the center of dilation.
- To find the coordinates of A', we need to move in the same direction from B by a distance of 2√5.
- Start from the coordinates of B (2,5) and move in the direction of A.
- Since the distance from B to A is √5, multiplying the direction vector by 2√5 will give us the coordinates of A'.
- The coordinates of A' are (2,5) + 2√5 * (unit direction vector from B to A).

5. Repeat steps 3 and 4 for the other vertices.
- Use the same direction and distance as A' for finding the coordinates of B' and C'.

Therefore, the vertices of triangle A'B'C' are:

A' = (2,5) + 2√5 * (unit direction vector from B to A),
B' = (2,5),
C' = (2,5) + 2√5 * (unit direction vector from B to C).