Question

For which of the following figures, diagonals are perpendicular to each other?

• A. Parallelogram
• B. Kite
• C. Trapezium
• D. Rectangle

Answer 1

The correct option is B

Kite

Explanation for the correct option:

A kite is defined as a parallelogram whose adjacent sides are equal.

In the given figure, consider the $△ABC$ and $△ADC$,

$BC=CD$ $[\because$Adjacent sides$]$

$AB=AD$ $[\because$Adjacent sides$]$

$AC=AC$ $[\because$Common side$]$

$\therefore △ABC\cong △ADC$ $[\because$SSS rule$]$

Consider $△BCO$ and $△DCO$,

$\angle BCO=\angle DCO$ $[\because$Corresponding sides of congruent triangles $△ABC$ and $△ADC$$]$

$BC=DC$ $[\because$Adjacent sides$]$

$CO=CO$ $[\because$Common side$]$

$\therefore △DCO\cong △BCO$ $[\because$SAS rule$]$

From the congruence of $△DCO$ and $△BCO$,

$\angle BOC=\angle DOC$

These angles are the only two angles on the line $DB$.

We know that the angle of a line is $180°$. Thus,

$\angle BOC+\angle DOC=180°$

Since the sum of angles $\angle BOC$ and $\angle DOC$ is $180°$ and that they are also equal, it must be that they measure $90°$.

This implies that line $AC$ and $DB$ are perpendicular.

Therefore, the diagonals of a kite are perpendicular to each other.

Hence, option B is correct.

Explanation for the incorrect options:

Option A:

1. A parallelogram is a quadrilateral whose opposite sides are equal and parallel.
2. Here, $\angle ABC\ne \angle DCB$ because adjacent sides are supplementary (because they have equal slopes, and are transverses of the same pair of parallel lines), it isn't necessary that they are equal.
   $\Rightarrow △ABC\ncong △DCB$
3. Consequently, $AC$ is not perpendicular to $BD$
4. Thus, diagonals are not perpendicular in a parallelogram.

Hence, option A is incorrect.

Option C:

1. A trapezium is a quadrilateral whose only one pair of opposite sides are parallel.
2. Here, since $AB\ne DC$ since it isn't necessary that the parallel sides are equal, the line segments joining them aren't equal.
   $\Rightarrow △ABC\ncong △DCB$
3. Consequently, $AC$ is not perpendicular to $BD$.
4. Thus, diagonals are not perpendicular in a trapezium.

Hence, option C is also incorrect.

Option D:

1. A rectangle is a parallelogram whose all angles are equal and measure $90°$.
2. Since we have already proven that a parallelogram's diagonals aren't perpendicular, it similarly holds true for a rectangle which is a type of a parallelogram.
3. Thus, diagonals are not perpendicular in a rectangle.

Hence, option D is also incorrect.

Therefore, the correct option is option B.