Question

Let R be the diamond region with vertices (0,2),(1,4),(2,2), and (1,0). Using Change of Variables to transform the region R into a rectangular region: - Find the area of the Diamond. - Find the area of the Transformed Diamond (rectangle) - What is the relationship between the Diamond and the Transformed Diamond?

Answer 1

The area of the diamond can be calculated by dividing it into two congruent triangles, finding the area of one, and multiplying by two. Transforming the diamond into a rectangle using a change of variable gives a rectangle with same area but different orientation. The relationship between the two shapes is that their areas are equivalent.

Step-by-step explanation:

To find the area of the diamond with vertices (0,2),(1,4),(2,2), and (1,0), we can split the diamond into two equal triangles and find the area of one. The base of each triangle is the length of one side of the diamond, which is the distance between (0,2) and (1,4) or (1,4) and (2,2). This can be calculated using the distance formula √((x2-x1)² + (y2-y1)²). The height of the triangles is the perpendicular distance from a vertex on the longer diagonal to the base, which is 2 units. Hence, the area of each triangle is 1/2 × base × height. Calculating the base and height, we find that the base is √5 and the height is 2. The area of one triangle is 1/2 × √5 × 2, and the area of the diamond is twice this value.

Using the Change of Variables technique, we can transform the diamond region into a rectangular region where the dimensions are aligned with the coordinates of the vertices. If the diamonds vertical diagonal is taken as the transformed rectangles length and the horizontal diagonal as its width, the rectangle would have the same area as the diamond since a rigid motion like rotation or translation does not change the region's area.

The relationship between the diamond and the transformed diamond is that they have the same area, but different orientations and shapes. The diamond has been reoriented such that its vertices lie on the coordinates in a way that makes it take on a rectangular shape.

Answer 2

Area of the Diamond: 6 square units

Area of the Transformed Rectangle: 6 square units

Relationship: The transformed rectangle is the same area as the original diamond, but with its vertices shifted and sides aligned with the coordinate axes.

Step-by-step explanation:

Area of the Diamond: Divide the diamond into two right triangles and find their areas:

Triangle 1: base = 1, height = 2. Area = (1/2)bh = 1

Triangle 2: base = 1, height = 2. Area = 1

Total area of the diamond: 1 + 1 = 2 square units

Area of the Transformed Rectangle: Using Change of Variables, the diamond is transformed into a rectangle with sides 2 and 3 (mapping the horizontal and vertical stretches of the diamond). Area of the rectangle: 2 * 3 = 6 square units

Relationship: Although the shapes differ in orientation and vertex positions, they maintain the same area. This is because the transformation stretches the diamond proportionally in both directions, preserving its overall size.

Therefore, the transformed rectangle has the same area as the original diamond, showcasing the flexibility of Change of Variables in geometric transformations.