Question

Triangle $ABC$ lies in the first quadrant. Points $A,$ $B,$ and $C$ are reflected across the line $y=x$ to points $A',$ $B',$ and $C',$ respectively. Assume that none of the vertices of the triangle lie on the line $y=x.$ Which of the following statements is not always true?

Triangle $A'B'C'$ lies in the first quadrant.

Triangles $ABC$ and $A'B'C'$ have the same area.

The slope of line $AA'$ is $-1.$

The slopes of lines $AA'$ and $CC'$ are the same.

Lines $AB$ and $A'B'$ are perpendicular to each other.

Answer: E

Concepts:transformation coordinate geometry slope

Difficulty rating: 1020

Solution:

Reflection across $y=x$ sends $(x,y)$ to $(y,x)$ . It therefore preserves the first quadrant and preserves area, so A and B are always true.

The direction from $(x,y)$ to its image $(y,x)$ is $(y-x,x-y)$ , whose slope is $-1$ because $x\ne y$ . Thus C and D are always true as well.

For E, take $A=(2,1)$ , $B=(3,2)$ , and $C=(4,1)$ . These points satisfy all the conditions, but $AB$ and $A'B'$ both have slope $1$ , so the two lines are parallel rather than perpendicular.

Thus, the answer is E .

2019 AMC 10B Problem 5

Answer: E

Difficulty rating: 1020

Solution:

Problem 5 in Other Years