Question

Given that ABCD is a parallelogram. What kind of quadrilateral is it if:
(a) AC=BD and AC is perpendicular to BD?
(b) AC is perpendicular to BD but is not equal to it?
(c) AC=BD but AC is not perpendicular to BD?

Answer 1

Hint:Here, in the given question, we are given that ABCD is a parallelogram (Parallelogram is a quadrilateral with 2 pairs of parallel opposite and equal sides). Here, we need to find which type of quadrilateral will be formed from the parallelogram if we apply some changes in it. We will solve each case one-by-one.

Complete step by step answer:
(a)

AC=BD (Given)
And $AC \bot BD$ (Given)
Diagonals of quadrilaterals are equal and they are perpendicular to each other. Thus, ABCD is a square.

(b)

$AC \bot BD$......(Given)
But AC and BD are not equal. Thus, ABCD is a rhombus.
(c)

AC=BD but AC and BD are not perpendicular to each other.
Thus, ABCD is a rectangle.

Note:To solve these types of questions, one must know all the properties of different quadrilaterals. Remember that,
-Square is a quadrilateral with four right angles ($90^\circ $). In a square, both pairs of opposite sides are parallel and equal in length. All sides of a square are congruent. Since all four sides of a square are the same length, that means there are two pairs of equal-length sides in a square.
-Rhombus is a quadrilateral with all four sides having equal lengths. The Opposite sides of a rhombus are equal and parallel, and the opposite angles are the same. It is a special case of parallelogram, whose diagonals intersect each other at $90^\circ $.
-Rectangle is a quadrilateral whose opposite sides are parallel and equal to each other. The diagonals bisect each other and both the diagonals have the same length.