Question

For which of the following figures, diagonals are equal?

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

Answer 1

The correct option is D

Rectangle

Explanation for the correct option:

For rectangles, diagonals are equal as shown in the figure

In the figure, consider $△ABC$ and $△DCB$

$\angle ABC=\angle DCB=90°$ $[\because$Angles of a rectangle$]$

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

$AB=DC$ $[\because$Opposite sides of a parallelogram are equal$]$

$\therefore △ABC\cong △DCB$

$\therefore AC=BD$ $[\because$Corresponding sides of congruent triangles$]$

Therefore, the diagonals of a rectangle are equal.

Hence, option D is correct.

Explanation for the incorrect options:

Option A:

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.
   $\begin{array}{cc}\Rightarrow & △ABC\ncong △DCB\\ \Rightarrow & AC\ne BD\end{array}$,

Hence, option A is incorrect.

Option B:

1. A rhombus is a quadrilateral whose opposite angles are equal.
2. Here, $\angle ABC\ne \angle DCB$ because they are adjacent sides.
   $\begin{array}{cc}\Rightarrow & △ABC\ncong △DCB\\ \Rightarrow & AC\ne BD\end{array}$

Hence, option B is incorrect.

Option C:

1. A rhombus is a quadrilateral whose opposite angles are equal.
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.
   $\begin{array}{cc}\Rightarrow & △ABC\ncong △DCB\\ \Rightarrow & AC\ne BD\end{array}$

Hence, option C is incorrect.

Therefore, option D is the correct option.