Question

How do you find the area of the pre image and the image?

Coordinates for the pre-image are X(3,18) Y(18,18) and Z(18,9) and the image coordinates are X’(1,6) Y’(6,6) Z’ (6,3)

and what is the relationship between the two areas?​

Answer 1

Answer: To find the area of the pre-image and the image, you can use the coordinates of the vertices of the shapes to calculate the lengths of the sides and use the formula for the area of a triangle:

Area = (base * height)/2

For the pre-image, the coordinates are X(3,18) Y(18,18) and Z(18,9). The base of the triangle is the distance between X and Z, which is 18 - 3 = 15. The height of the triangle is the distance between Y and Z, which is 18 - 9 = 9. The area of the pre-image is therefore:

Area = (15 * 9)/2 = 67.5

For the image, the coordinates are X’(1,6) Y’(6,6) Z’ (6,3). The base of the triangle is the distance between X’ and Z’, which is 6 - 1 = 5. The height of the triangle is the distance between Y’ and Z’, which is 6 - 3 = 3. The area of the image is therefore:

Area = (5 * 3)/2 = 7.5

The relationship between the two areas is that the area of the image is smaller than the area of the pre-image. This is because the image has been scaled down in size relative to the pre-image. The amount by which the image has been scaled down can be calculated by dividing the area of the image by the area of the pre-image:

Scale factor = area of image / area of pre-image = 7.5 / 67.5 = 0.11

This means that the image has been scaled down by a factor of 0.11 relative to the pre-image.

Explanation:

Answer 2

• preimage area: 67.5 square units
• image area: 7.5 square units
• image area is 1/9 of preimage area, the square of the scale factor

Explanation:

You want to know the areas and their relationship of the preimage with coordinates X(3,18), Y(18,18), and Z(18,9), and the image with coordinates X’(1,6), Y’(6,6), Z’ (6,3).

Triangles

When you plot the points on a graph, you see they define right triangles. Each has sides aligned with the x- and y-axes, so it is easy to find their dimensions by counting grid squares or subtracting coordinates.

The preimage triangle is 18-3 = 15 units in the x-direction, and 18-9 = 9 units in the y-direction.

The image triangle is 6-1 = 5 units in the x-direction, and 6-3 = 3 units in the y-direction.

We notice that both the coordinates and the dimensions of the image triangle are 1/3 those of the preimage triangle.

Areas

The area of each triangle is given by the formula ...

A = 1/2bh

Using the dimensions we found above, the areas are ...

preimage area: 1/2(15)(9) = 135/2 = 67.5 . . . . square units

image area: 1/2(5)(3) = 15/2 = 7.5 . . . . square units

Relationship

We have observed that the image dimensions are 1/3 those of the preimage. The image area is 7.5/67.5 = 1/9 of that of the preimage. This value is the square of the scale factor of the dimensions,