Question

Which of the following statement is true for a parallelogram :

• A. Its diagonals are equal
• B. Its diagonals are perpendicular to each other
• C. The diagonals divide the parallelogram into four congruent triangles
• D. The diagonals bisect each other

Hint:

Key properties of parallelogram diagonals:
{Always true:} Diagonals bisect each other.
{Sometimes true (special cases):}
Diagonals are equal (only in rectangles and squares).
Diagonals are perpendicular (only in rhombuses and squares).
Diagonals divide into four congruent triangles (only in rhombuses and squares). The question asks for a statement true for {a} parallelogram (meaning any parallelogram).

Answer 1

The Correct Option is D

Concept: A parallelogram is a quadrilateral with two pairs of parallel sides. It has several important properties related to its sides, angles, and diagonals. Step 1: Analyzing the properties of a parallelogram's diagonals Let's examine each statement:
(1) Its diagonals are equal: This is not always true for a general parallelogram. Diagonals are equal only in special types of parallelograms, such as rectangles and squares. For a general parallelogram (like a rhombus that is not a square, or a typical "slanted" parallelogram), the diagonals are of different lengths.
(2) Its diagonals are perpendicular to each other: This is not always true for a general parallelogram. Diagonals are perpendicular to each other only in special types of parallelograms, such as rhombuses and squares.
(3) The diagonals divide the parallelogram into four congruent triangles: This is not true. The two diagonals divide the parallelogram into four triangles. The triangles formed by opposite vertices and the intersection point are congruent in pairs (e.g., \(\triangle \text{AOB} \cong \triangle \text{COD}\) and \(\triangle \text{BOC} \cong \triangle \text{DOA}\) if O is the intersection). However, all four are generally not congruent to each other unless it's a special case like a rhombus (where all four are congruent right-angled triangles) or a rectangle (where they are congruent in pairs of isosceles triangles, but not all four are congruent unless it's a square). The statement says "four congruent triangles," which is only true for a rhombus (and hence a square). A diagonal divides a parallelogram into two congruent triangles.
(4) The diagonals bisect each other: This is a fundamental property true for all parallelograms. "Bisect" means they cut each other into two equal halves at their point of intersection. If O is the intersection of diagonals AC and BD, then AO = OC and BO = OD. Step 2: Identifying the universally true statement The property that holds true for every parallelogram, regardless of whether it's a rectangle, rhombus, square, or just a general parallelogram, is that its diagonals bisect each other. Therefore, the correct statement is that the diagonals bisect each other.