Question

Show that the diagonals of quadrilateral are perpendicular.

Short answer

Hence, the diagonals of the quadrilateral are perpendicular to each other.

Step-by-step solution

Draw a quadrilateral with same lengths on both sides.

Draw a quadrilateral with same lengths on both sides as shown in Figure 1.

Use the Figure 1 for explanation of the solution.

From Figure 1, the diagonal vectors of quadrilateral are \(\overrightarrow {AC} \)and \(\overrightarrow {BD} \)

Find the vector\(\overrightarrow {AC} \) from Figure 1.

\(\overrightarrow {AC} = \overrightarrow {AB} + \overrightarrow {BC} \)

Find the vector\(\overrightarrow {BD} \) from Figure 1.

\(\overrightarrow {BD} = \overrightarrow {BC} + \overrightarrow {CD} \)

Here, \(\overrightarrow {CD} \) has equivalent magnitude of\(\overrightarrow {AB} \) with opposite direction. So that the sum\(\overrightarrow {CD} \) can be defined as follows.

\(\overrightarrow {CD} = - \overrightarrow {AB} \)

Find dot product between vectors\(\overrightarrow {AC} \)and\(\overrightarrow {BD} \).

Expand this expression by using the formula\((a + b)(a - b) = {a^2} - {b^2}.\)

\(\overrightarrow {AC} \cdot \overrightarrow {BD} = |\overrightarrow {BC} {|^2} - |\overrightarrow {AB} {|^2}\)

As length of all sides are equal, the magnitude of\(|\overrightarrow {BC} |\) and\(|\overrightarrow {AB} |\) are equal.

Substitute\(|\overrightarrow {AB} |\)for\(|\overrightarrow {BC} |\).

\(\begin{aligned}{c}\overrightarrow {AC} \cdot \overrightarrow {BD} &= |\overrightarrow {AB} {|^2} - |\overrightarrow {AB} {|^2}\\ &= 0\end{aligned}\)

The dot product between two line vectors\(\overrightarrow {AC} \) and\(\overrightarrow {BD} \) becomes zero that is it satisfies the condition of the perpendicular vector.

Hence, the diagonals of the quadrilateral are perpendicular to each other.