ABCD IS A SQUARE . L IS ANY LINE IN ITS PLANE. SHOW THAT THE SUM OF THE SQUARED PROJECTIONS OF THE SIDES OF SQUARE IS INDEPENDENT OF L

To tackle the problem of showing that the sum of the squared projections of the sides of a square onto any line in its plane is independent of that line, let's break it down step by step. We’ll use some geometry and vector projections to illustrate the concept clearly.

Understanding the Setup

Consider a square ABCD with vertices A(0, 0), B(1, 0), C(1, 1), and D(0, 1). The sides of the square are AB, BC, CD, and DA. We want to analyze how these sides project onto an arbitrary line L in the plane.

Defining the Line L

Let’s define line L by its direction vector. We can express this vector as v = (cos θ, sin θ), where θ is the angle that line L makes with the positive x-axis. This representation allows us to consider any line in the plane by varying θ.

Calculating Projections

The projection of a vector onto another vector can be calculated using the formula:

proj_v(u) = (u · v) / (v · v) * v

For our square, the sides can be represented as vectors:

• AB = (1, 0)
• BC = (0, 1)
• CD = (-1, 0)
• DA = (0, -1)

Finding Squared Projections

Now, let’s compute the projections of each side onto the line L:

• For side AB:

  proj_L(AB) = ((1, 0) · (cos θ, sin θ)) * (cos θ, sin θ) = (cos θ, 0)

  Squared projection = |proj_L(AB)|² = (cos² θ)

• For side BC:

  proj_L(BC) = ((0, 1) · (cos θ, sin θ)) * (cos θ, sin θ) = (0, sin θ)

  Squared projection = |proj_L(BC)|² = (sin² θ)

• For side CD:

  proj_L(CD) = ((-1, 0) · (cos θ, sin θ)) * (cos θ, sin θ) = (-cos θ, 0)

  Squared projection = |proj_L(CD)|² = (cos² θ)

• For side DA:

  proj_L(DA) = ((0, -1) · (cos θ, sin θ)) * (cos θ, sin θ) = (0, -sin θ)

  Squared projection = |proj_L(DA)|² = (sin² θ)

Summing the Squared Projections

Now, let’s sum the squared projections of all four sides:

Sum = cos² θ + sin² θ + cos² θ + sin² θ = 2(cos² θ + sin² θ)

Using the Pythagorean Identity

According to the Pythagorean identity, we know that:

cos² θ + sin² θ = 1

Thus, we can simplify our sum:

Sum = 2(1) = 2

Final Thoughts

This result shows that the sum of the squared projections of the sides of the square onto any line L is always equal to 2, regardless of the orientation of the line. Therefore, we conclude that this sum is indeed independent of the line L. This property highlights a fascinating aspect of geometry, where certain relationships remain constant despite changes in perspective.