Tomika heard that the diagonals of a rhombus are perpendicular to each other. Help her test her conjecture. Graph quadrilateral $ABCD$ on a coordinate grid with $A(1, 4)$, $B(6, 6)$, $C(4, 1)$, and $D(–1, –1)$.   

1. A rhombus has four sides that are equal length. Is $ABCD$ a rhombus? Show how you know.

   Step 1(a):

   Create slope triangles on each side. Visualize them using the eTool, then draw them on your paper.

   Step 2(a):

   Use the Pythagorean Theorem to calculate the length of each side. Are they the same? If so, the shape is a rhombus.

2. Write the equations of the lines on which the diagonals lie. That is, write the equations of $\overleftrightarrow{ A C }$and $\overleftrightarrow{ B D }$.

   Hint (b):

   Connect points $A$ and $C$ and points $B$ and $D$. Use slope-intercept form ($y = mx + b$) to write the equations of the lines. Test your equations in the eTool to be sure they are correct.

3. Compare the slopes of $\overleftrightarrow{ A C }$ and $\overleftrightarrow{ B D }$. What do you notice?

   Hint (c):

   What is true about the slopes of parallel and perpendicular lines?