Theorem Diagonals of a Rhombus are Perpendicular to each other Video Lecture | Mathematics (Maths) Class 9

Ans. The diagonals of a rhombus are perpendicular to each other because a rhombus is a special type of parallelogram where all sides are equal in length. This means that opposite angles in a rhombus are congruent. When the diagonals intersect, they form four congruent right triangles, where the hypotenuse of each triangle is a diagonal. Therefore, the diagonals of a rhombus are perpendicular to each other.

Ans. To prove that the diagonals of a rhombus are perpendicular, we can use the properties of a rhombus. Let's assume ABCD is a rhombus, where AC and BD are the diagonals. - Since ABCD is a rhombus, all sides are equal in length. This implies that AB = BC = CD = DA. - In a rhombus, opposite angles are congruent. Therefore, angle B = angle D. - By the Side-Angle-Side (SAS) congruence criterion, we can prove that triangle ABC is congruent to triangle CDA. - As a result, angle A = angle C. - Since angle B = angle D and angle A = angle C, we can conclude that angle A + angle B + angle C + angle D = 360 degrees. - In a quadrilateral, the sum of all angles is 360 degrees. Thus, angle A + angle B + angle C + angle D = 360 degrees. - Since angle A + angle B + angle C + angle D = 360 degrees and angle A = angle C, we can rewrite the equation as 2(angle A + angle B) = 360 degrees. - Simplifying the equation, we get angle A + angle B = 180 degrees. - This implies that angle A and angle B are supplementary angles. - Supplementary angles are always perpendicular to each other. - Therefore, the diagonals AC and BD of the rhombus ABCD are perpendicular to each other.

Ans. No, a rhombus cannot have diagonals that are not perpendicular to each other. By definition, a rhombus is a parallelogram with four sides of equal length. In a rhombus, opposite angles are congruent, and the sum of all angles is 360 degrees. When the diagonals of a rhombus intersect, they form four congruent right triangles, where the hypotenuse of each triangle is a diagonal. Since right angles are always perpendicular, the diagonals of a rhombus must be perpendicular to each other.

Ans. Yes, the diagonals of a rectangle are perpendicular to each other. A rectangle is a special type of parallelogram with four right angles. When the diagonals of a rectangle intersect, they form four congruent right triangles, where the hypotenuse of each triangle is a diagonal. Since right angles are always perpendicular, the diagonals of a rectangle must be perpendicular to each other.

Ans. Yes, the diagonals of a square are perpendicular to each other. A square is a special type of rhombus and rectangle, where all sides are equal in length and all angles are right angles. When the diagonals of a square intersect, they form four congruent right triangles, where the hypotenuse of each triangle is a diagonal. Since right angles are always perpendicular, the diagonals of a square must be perpendicular to each other.