Animated Solution for Mathematics - Straight Lines: The diagonals of a parallelogram $PQRS$ are along the lines $x+3y=4$ and $6x-2y=7$. Then $PQRS$ must be a

Imagine a parallelogram $PQRS$ floating in the Cartesian plane. We are given two lines, $L_1:x+3y=4$ and $L_2:6x-2y=7$, which represent the diagonals.

The beauty of coordinate geometry lies in the fact that we do not need to draw the parallelogram to understand its nature. We only need to understand the relationship between these two lines. The key to unlocking the identity of $PQRS$ lies in the angle at which these diagonals intersect.

Finding the Slopes

To find the angle between two lines, we must first find their slopes. Let us take the first diagonal, $L_1$, given by the equation $x+3y=4$.

To find the slope, we rearrange this into the slope-intercept form, $y=mx+c$. By isolating $y$, we get:

Thus, the slope $m_1$ is $-\frac{1}{3}$.

Now, let us turn our attention to the second diagonal, $L_2$, defined by $6x-2y=7$. Following the same logic, we rearrange to get:

Here, the slope $m_2$ is $3$.

The Product of Slopes

Now, we arrive at the moment of truth. We have $m_1=-\frac{1}{3}$ and $m_2=3$. Let us calculate their product:

This is not a coincidence; it is a mathematical signature. In coordinate geometry, when the product of the slopes of two lines is exactly $-1$, those lines are perpendicular, or $L_1\perp L_2$.

Why it is a Rhombus

We have discovered that the diagonals of our parallelogram $PQRS$ intersect at a $90^{\circ }$ angle. This is the defining property of a rhombus.

While all squares are rhombi, not all rhombi are squares. Because we lack information about the lengths of the diagonals, we cannot claim it is a square, but we can state with absolute certainty that $PQRS$ is a rhombus.

This problem teaches us that geometry is not about memorizing shapes, but about understanding the relationships between lines. Keep this logic in your toolkit, and you will find that even the most complex problems become simple, elegant stories.