Question

Geometry For any parallelogram, prove that the sum of the squares of the lengths of the sides equals the sum of the squares of the lengths of the diagonals.

Short answer

In a parallelogram, the sum of the squares of the sides equals the sum of the squares of the diagonals.

Step-by-step solution

Understand the Parallelogram Properties

A parallelogram has opposite sides that are equal and parallel. Opposite angles are also equal. Label the vertices as A, B, C, and D.

Define the Side Lengths

Let the lengths of sides AB and CD be represented as a, and the lengths of sides AD and BC be represented as b.

Define the Diagonals

Let the lengths of the diagonals AC and BD be represented as p and q respectively.

Apply the Parallelogram Law

According to the parallelogram law, for any parallelogram, the square of the length of a diagonal is equal to the sum of the squares of the lengths of two adjacent sides minus twice the product of their lengths times the cosine of the angle between them. Specifically, \[ AC^2 = AD^2 + AB^2 - 2 \times AD \times AB \times \text{cos}(\theta), \] where \( \theta \) is the angle between AD and AB.

Calculate Each Diagonal Using the Parallelogram Law

For diagonal AC: \[ p^2 = a^2 + b^2 + 2 \times ab \times \text{cos}(\theta). \]For diagonal BD: \[ q^2 = a^2 + b^2 - 2 \times ab \times \text{cos}(\theta). \]

Add the Squares of the Diagonals

Sum the squares of p and q to get: \[ p^2 + q^2 = (a^2 + b^2 + 2 \times ab \times \text{cos}(\theta)) + (a^2 + b^2 - 2 \times ab \times \text{cos}(\theta)). \]

Simplify the Expression

Notice that the terms involving \( \text{cos}(\theta) \) cancel each other out: \[ p^2 + q^2 = a^2 + b^2 + a^2 + b^2 = 2a^2 + 2b^2. \]

Conclusion

The sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides doubled: \[ 2a^2 + 2b^2. \]

Key concepts

Parallelogram Law

The Parallelogram Law is a fundamental concept in geometry. It states that the square of the length of a diagonal in a parallelogram equals the sum of the squares of the lengths of two adjacent sides, plus or minus twice the product of their lengths, and the cosine of the included angle between them.
In a parallelogram labeled ABCD, with AB = a and AD = b, we can define the diagonals AC and BD. According to the Parallelogram Law:
AC^2 = AB^2 + AD^2 - 2 AB * AD * cos(θ)
and
BD^2 = AB^2 + AD^2 - 2 AD * AB * cos(θ)
Here, θ represents the angle between AB and AD. This law is crucial for solving and proving properties related to parallelograms in geometry.

Diagonal Lengths

Understanding the lengths of the diagonals in a parallelogram is essential for many geometric proofs and calculations. Given a parallelogram with side lengths a and b, and diagonals labeled p (diagonal AC) and q (diagonal BD), you can use the Parallelogram Law to find their lengths:
p^2 = a^2 + b^2 + 2ab * cos(θ)
and
q^2 = a^2 + b^2 - 2ab * cos(θ)
When you add these two equations together, the terms involving the cosine function cancel out, making it easier to arrive at the final expression during proofs.

Trigonometric Identities

In the context of parallelograms, trigonometric identities such as cosine are pivotal. Cosine in particular helps relate the angles within the shape to its side lengths and diagonals.
Cosine values of the included angle θ between sides can be crucial for computing diagonal lengths accurately. For instance, in the formulas:
2ab * cos(θ)
otes how the trigonometric identity ties into the sides and angles. These identities simplify solving complex geometric problems, transforming them into manageable algebraic equations.

Geometric Properties

A parallelogram is defined by several key geometric properties. These properties lay the groundwork for various proofs:

• Opposite sides are equal and parallel.
• Opposite angles are equal.
• Adjacent angles sum up to 180 degrees.
• Diagonals bisect each other.

Understanding these properties helps visualize problems and simplify geometric calculations.
For example, recognizing that opposite sides are equal allows the use of side lengths a and b consistently throughout equations, whilst knowing diagonals bisect each other can simplify mid-point related computations.

Proofs in Geometry

Proofs are a structured way to demonstrate why geometric properties and theorems hold true. In the context of parallelograms, proofs might involve showing relationships between side lengths, angles, and diagonals using established theorems.
For instance, proving that the sum of squares of the diagonals equals the sum of the squares of the sides involves:

• Defining side lengths and diagonals.
• Applying the Parallelogram Law.
• Adding the squared lengths of the diagonals.
• Simplifying using trigonometric values.

Each step logically follows the previous, piecing together a clear conclusion, enhancing mathematical understanding and problem-solving skills.