Final answer:
Upon reflecting the triangle across the line x=8 and calculating the area of one triangle using the Shoelace formula, the area of the union is found to be twice that of one triangle, totaling 36 square units.
Step-by-step explanation:
To find the area of the union of two triangles when one is reflected about the line x=8, we need to consider the effects of reflection. Upon reflection across x=8, the x-coordinates of the vertices will change, but the y-coordinates will remain the same. To find the reflected points, subtract the original x-coordinate from twice the line of reflection (16). For example, reflecting point (6, 5) gives us (16-6, 5) which is (10, 5). When we reflect all points, we get the vertices of the second triangle: (10, 5), (8, -3), and (7, 1).
Since the original and reflected triangles do not overlap, we can find the area of each triangle individually and sum them up to get the total area of the union. To calculate the area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3), we use the Shoelace formula: Area = |(x1y2 + x2y3 + x3y1) - (x2y1 + x3y2 + x1y3)| / 2.
Applying this formula to the original triangle and to the reflected triangle, we get the same area for both, as they are congruent. Thus, we only need to calculate the area for one triangle and multiply by two. After calculation, the area of one triangle, say the original one, is found to be 18 square units. Therefore, the union of the two triangles would have an area of 2 × 18 = 36 square units.