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.

Solution:

Choice A must be true since the reflection of the first quadrant is itself, so anything inside stays inside after a reflection.

Choice B must be true as a reflection keeps the same area.

Choice C must be true as a reflection will have a perpendicular slope to the line its reflected about, so its slope is $-\frac 11 = -1.$

Choice D must be true as they both have the slope of $-1.$

Choice E can be false as if $A = (2,1),$ $B=(3,2),$ then $AB$ and its reflection both have slope $-1,$ making them parallel. Therefore, they can be not perpendicular.

Thus, the answer is E .

2019 AMC 10B Problem 5

Solution:

Problem 5 in Other Years