SAT Mathematics Practice Test 2

In the xy-plane, the slope m of the line that passes through the points $\left({\mathit{x}}_1,{\mathit{y}}_1\right)$ and $\left({\mathit{x}}_2,{\mathit{y}}_2\right)$ is $\mathit{m}=\frac{{\mathit{y}}_2-{\mathit{y}}_1}{{\mathit{x}}_2-{\mathit{x}}_{\mathit{i}}}$ Thus, the slope of the line through the points $\mathit{C}\left(7,2\right)$ and $\mathit{E}\left(1,0\right)$ is $\frac{2-0}{7-1},$which simplifies to $\frac{2}{6}=\frac{1}{3}$.Therefore, diagonal $\mathit{A}\mathit{C}$ has a slope of $\frac{1}{3}$.The other diagonal of the square is a segment of the line that passes through points $\mathit{B}$ and $\mathit{D}$. The diagonals of a square are perpendicular, and so the product of the slopes of the diagonals is equal to $-1$. Thus, the slope of the line that passes through $\mathit{B}$ and $\mathit{D}$ is $-3$ because $\frac{1}{3}\left(-3\right)$.Hence an equation of the line that passes through $\mathit{B}$ and $\mathit{D}$ can be written as $\mathit{y}=-3\mathit{x}+\mathit{b}$, where $\mathit{b}$ is the y-intercept of the line. Since diagonal $\mathit{B}\mathit{D}$ will pass through the center of the square, $\mathit{E}\left(1,0\right)$,the equation $0=-3\left(1\right)+\mathit{b}$ holds. Solving this equation for b gives $\mathit{b}=3$. Therefore, an equation of the line that passes through points B and D is $\mathit{y}=-3\mathit{x}+3$,which can be rewritten as $\mathit{y}=-3\left(\mathit{x}-1\right)$
Choices A, C, and D are incorrect and may result from a conceptual error or a calculation error.