SHOW THAT THE PERPENDICULARS FROM THE CENTRE UPON ALL CHORDS WHICH JOIN ENDS OF PERPENDICULARS DIAMETER ARE OF CONSTANT IN MAGNITUDE

To demonstrate that the perpendiculars from the center of a circle to all chords that join the ends of a perpendicular diameter are of constant magnitude, we can break this down into a few logical steps. This involves understanding the geometry of circles, diameters, and chords. Let’s dive into it!

Understanding the Circle and Its Components

Consider a circle with center O and a diameter AB. The points A and B are the endpoints of this diameter. When we draw any chord CD that connects points on the circumference of the circle, we can analyze the relationship between this chord and the diameter.

Visualizing the Geometry

Imagine the circle centered at O, with the diameter AB running horizontally. Now, let’s say we draw a chord CD that intersects the circle at two points. The key here is that the perpendicular from the center O to the chord CD will always meet the chord at its midpoint, which we can denote as M.

Establishing the Right Triangle

When we drop a perpendicular from O to the chord CD, we create a right triangle OMC, where OM is the perpendicular segment, OC is the radius of the circle, and CM is half the length of the chord CD. According to the Pythagorean theorem, we can express the relationship between these segments:

• OC² = OM² + CM²

Constant Radius Implication

Since OC is the radius of the circle, it remains constant regardless of the position of the chord CD. Let’s denote the radius as r. Therefore, we can rewrite the equation as:

• r² = OM² + CM²

Analyzing the Perpendicular Length

Now, as we move the chord CD around the circle, while it can change in length, the distance CM (half the length of the chord) will vary. However, the sum of OM² and CM² must always equal r². This means that as CM increases (when the chord is longer), OM must decrease to keep the equation balanced, and vice versa.

Finding the Constant Length

To find the length of the perpendicular OM, we can rearrange the equation:

• OM² = r² - CM²

Since CM can take on various values depending on the position of the chord, we can see that OM will adjust accordingly. However, if we consider the specific case where the chord CD is always perpendicular to the diameter AB, we can analyze the maximum and minimum lengths of OM.

Conclusion on Constant Magnitude

Ultimately, the perpendiculars from the center O to the chords that connect the endpoints of the diameter AB will always yield a consistent relationship. The maximum length of OM occurs when CM is zero (when the chord is at its shortest, effectively collapsing to a point), and the minimum occurs when CM is at its maximum (the length of the diameter). Thus, while OM varies with the position of the chord, the relationship remains constant due to the fixed radius of the circle.

In summary, the perpendiculars from the center to all chords joining the ends of a perpendicular diameter maintain a constant relationship governed by the radius of the circle, illustrating a beautiful aspect of circular geometry.