3.3.6. Diagonals of a parallelogram

The diagonals of a parallelogram, in general, are not of equal length. However, the diagonals of a parallelogram have an interesting property.

In a parallelogram, while the diagonals do not necessarily have the same length, they do intersect in a way that is consistent across all parallelograms. This means that the two diagonals divide each other into two equal parts. This property is crucial for understanding the relationship between the sides of a parallelogram and provides insights into its geometry.

Think of a playground seesaw. The point where the seesaw balances is like the intersection of the diagonals in a parallelogram. No matter how heavy one end is, the seesaw balances at its midpoint. Similarly, in a parallelogram, the diagonals meet at a point that equally divides them, showing balance.

DO THIS: Take a cut-out of a parallelogram, say, ABCD (Fig 3.23). Let its diagonals AC and DB meet at O. Find the mid point of AC by a fold, placing C on A. Is the mid-point the same as O? Does this show that diagonal DB bisects the diagonal AC at the point O?

This hands-on activity demonstrates the property of diagonal bisection. By folding the diagonal AC, students can visibly see where the midpoint is located. When they place point C directly over point A, if the midpoint aligns with point O (the intersection point), it confirms that diagonal DB bisects AC at that point. If this observation is repeated with diagonal DB, it further emphasizes the bisection property.

Imagine cutting a pizza in half. When you slice it, no matter how you cut, the center point of the pizza is the same. This is much like how the diagonals of a parallelogram cut through each other; they divide each other evenly at their intersection point, just like the pizza is split evenly at the center.

To argue and justify this property is not very difficult. From Fig 3.24, applying ASA criterion, it is easy to see that ∆ AOB ≅ ∆ COD (How is ASA used here?). This gives AO = CO and BO = DO.

The ASA (Angle-Side-Angle) criterion states that if two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, then the two triangles are congruent. In the context of diagonals in a parallelogram, we establish that triangles AOB and COD have equal angles and a shared side (the line segment that forms part of both triangles). Consequently, this congruence allows us to conclude that AO equals CO and BO equals DO—demonstrating that the diagonals bisect each other.

Think about a kite’s tail. The two equal-length lines representing the tail show how each part splits evenly, creating symmetry. Much like the diagonals in a parallelogram creating symmetrical triangles at their intersection, this symmetry showcases balance and equality.

Example 6: In Fig 3.25 HELP is a parallelogram. (Lengths are in cms). Given that OE = 4 and HL is 5 more than PE? Find OH.

In this example, we apply the property of diagonal bisection to calculate lengths in the parallelogram. Knowing that OE equals 4 allows us to immediately conclude that OP is also 4, as diagonals bisect each other. Since we can also say that PE equals 8 (given OH plus OE), we can deduce all related lengths. Solving the equations built around these relationships allows students to apply theory to practice.

Consider using a ruler to measure the lengths of a shelf. Just like each segment of the shelf needs to be balanced and measured accurately, the diagonals in a parallelogram maintain balance and equal division, allowing for straightforward calculations of different parts based on established properties.

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