2019 AMC 10B Problems/Problem 5
Triangle lies in the first quadrant. Points , , and are reflected across the line to points , , and , respectively. Assume that none of the vertices of the triangle lie on the line . Which of the following statements is not always true?
Triangle lies in the first quadrant.
Triangles and have the same area.
The slope of line is .
The slopes of lines and are the same.
Lines and are perpendicular to each other.
Let's analyze all of the options separately.
A: Clearly A is true, because a coordinate in the first quadrant will have (+,+), and its inverse would also have (+,+)
B: The triangles have the same area, it's the same triangle.
C: If coordinate A has (x,y), then its inverse will have (y,x). (x-y)/(y-x)=-1, so this is true.
D: Likewise, if coordinate A has (x1,y1), and AA' has a slope of -1, then coordinate B, with (x2,y2), will also have a slope of -1. This is true.
E: By process of elimination, this is the answer, but if coordinate A has (x1,y1) and coordinate B has (x2,y2), then their inverses will be (y1,x1), (y2,x2), and it is not necessarily true that (y2-y1)/(x2-x1)=-(y2-y1)/(x2-x1). (Negative inverses of each other).
Clearly, the answer is .
If and , then the slope of , , is , while the slope of , , is . is the of , but it is not the negative reciprocal of . To generalize, let denote the coordinates of point A, let denote the coordinates of point B, let denote the slope of segment , and let denote the slope of segment . Then, the coordinate pair for is , and the pair for is . Then, , and . If and , , and in these cases, the condition is false.
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