From A Point On A Circle, Two Perpendicular Chords Are Drawn, Which Are Located At A Distance Of 6 Cm And 11 Cm From The Center Of The Circle. Find The Lengths Of These Chords.

Jun 21, 2025 by ADMIN 177 views

Finding Chords in a Circle A Geometry Challenge

Introduction

In the realm of geometry, circles hold a special allure, their perfect symmetry and elegant properties captivating mathematicians and enthusiasts alike. Within the embrace of a circle, chords, those line segments connecting two points on the circumference, play a crucial role. When these chords intertwine in specific ways, such as forming perpendicular intersections, geometric puzzles arise, challenging our spatial reasoning and problem-solving skills. In this article, we will delve into one such puzzle, where two mutually perpendicular chords emanate from a point on a circle, their distances from the circle's center known. Our mission is to unravel the lengths of these enigmatic chords.

Problem Statement: Unveiling the Secrets of Perpendicular Chords

Imagine a circle, a timeless symbol of completeness. From a point gracefully perched on its circumference, two chords emerge, their paths intersecting at a perfect right angle. These mutually perpendicular chords dance within the circle's embrace, their positions further defined by their distances from the circle's center. One chord stands 6 cm away from the heart of the circle, while its perpendicular counterpart lies 11 cm distant. The challenge that beckons us is to determine the lengths of these two intriguing chords.

Visualizing the Geometry

Before we embark on the mathematical journey, let's paint a mental picture of the scenario. Envision the circle, its center a beacon of symmetry. The two perpendicular chords, like outstretched arms, reach across the circle's interior. Their distances from the center act as guiding threads, anchoring their positions within the circular domain. To solve this puzzle, we must bridge the gap between these distances and the chords' elusive lengths.

Deconstructing the Problem: A Strategic Approach

To conquer this geometric challenge, we'll employ a strategic approach, breaking down the problem into manageable steps. First, we'll introduce key geometric concepts and theorems that will serve as our tools. Then, we'll carefully construct a diagram, a visual representation of the problem that will illuminate the relationships between the circle, its center, the chords, and their distances. Finally, we'll wield our geometric tools, applying theorems and principles to derive equations that will ultimately reveal the lengths of the chords.

Geometric Arsenal: Essential Tools for the Quest

To navigate the intricacies of this problem, we'll need to equip ourselves with a set of geometric tools, fundamental concepts and theorems that will guide our way.

The Circle's Embrace: Radius, Diameter, and Center

At the heart of our problem lies the circle, a geometric entity defined by its radius, the distance from the center to any point on the circumference. The diameter, a line segment passing through the center and connecting two points on the circle, is twice the radius. The center, the circle's nucleus, serves as a point of symmetry, anchoring the circle's form.

Chords: Lines Within the Circle

A chord, as we've mentioned, is a line segment that bridges two points on the circle's circumference. The longest chord, the one that passes through the center, is the diameter. Chords hold a unique relationship with the circle's center, their distances from the center influencing their lengths.

Perpendicularity: Right Angles and Harmony

Perpendicularity, the essence of our problem, signifies the intersection of two lines or segments at a right angle, a perfect 90-degree corner. Perpendicular lines create a sense of balance and harmony, their right angles forming the foundation for many geometric constructions.

The Pythagorean Theorem: A Cornerstone of Geometry

The Pythagorean Theorem, a timeless cornerstone of geometry, unveils a profound relationship between the sides of a right triangle. In a right triangle, the square of the hypotenuse, the side opposite the right angle, equals the sum of the squares of the other two sides, the legs. This theorem will prove invaluable in connecting the distances of the chords from the center to their lengths.

The Perpendicular Bisector Theorem: A Bridge to the Center

The Perpendicular Bisector Theorem states that the perpendicular bisector of a chord, a line that intersects the chord at its midpoint and forms a right angle, always passes through the center of the circle. This theorem will help us establish a crucial link between the chords and the circle's center.

Constructing the Diagram: A Visual Roadmap

With our geometric tools in hand, let's construct a diagram, a visual roadmap that will guide our solution. Draw a circle, its center marked as O. Now, sketch two perpendicular chords, AB and CD, emanating from a point P on the circle's circumference. Let the distances from the center O to the chords AB and CD be 6 cm and 11 cm, respectively. Label the midpoints of AB and CD as M and N, respectively. This diagram, our visual compass, will illuminate the relationships between the elements of the problem.

Unveiling the Right Triangles: A Geometric Revelation

As we gaze upon our diagram, right triangles emerge, geometric structures that hold the key to our solution. Triangle OMA, formed by the center O, the midpoint M of chord AB, and endpoint A, is a right triangle. Similarly, triangle ONC, formed by the center O, the midpoint N of chord CD, and endpoint C, is also a right triangle. These right triangles, governed by the Pythagorean Theorem, will allow us to connect the distances of the chords from the center to their lengths.

Applying the Theorems: Unraveling the Equations

Now, let's wield our geometric tools, applying the Pythagorean Theorem and the Perpendicular Bisector Theorem to derive equations that will reveal the lengths of the chords.

Connecting Distances and Lengths: The Pythagorean Theorem in Action

In right triangle OMA, the Pythagorean Theorem dictates that OA² = OM² + AM². Here, OA is the radius of the circle, OM is the distance from the center to chord AB (6 cm), and AM is half the length of chord AB. Similarly, in right triangle ONC, we have OC² = ON² + NC², where OC is the radius, ON is the distance from the center to chord CD (11 cm), and NC is half the length of chord CD.

The Perpendicular Bisector Theorem: A Bridge to the Center

The Perpendicular Bisector Theorem assures us that lines OM and ON, the perpendicular bisectors of chords AB and CD, respectively, pass through the center O. This theorem confirms our diagram's construction and reinforces the validity of our approach.

Establishing the Equations: A Mathematical Formulation

Let the radius of the circle be r, the length of chord AB be x, and the length of chord CD be y. From the Pythagorean Theorem applied to triangles OMA and ONC, we have:

r² = 6² + (x/2)² r² = 11² + (y/2)²

A Crucial Insight: Connecting the Chords

Here's where a crucial insight comes into play. Since chords AB and CD are perpendicular, quadrilateral OMPN is a rectangle. This implies that OM² + ON² = OP², where OP is the radius r. Therefore, we have:

6² + 11² = r²

Solving the Equations: Unveiling the Chord Lengths

Now, we have a system of equations that we can solve to find the lengths of the chords.

Determining the Radius: The Circle's Measure

From the equation 6² + 11² = r², we find r² = 36 + 121 = 157. Therefore, the radius r = √157 cm.

Finding the Chord Lengths: The Final Revelation

Substituting r² = 157 into the equations r² = 6² + (x/2)² and r² = 11² + (y/2)², we get:

157 = 36 + (x/2)² 157 = 121 + (y/2)²

Solving for x and y, we find:

(x/2)² = 121, so x/2 = 11, and x = 22 cm (y/2)² = 36, so y/2 = 6, and y = 12 cm

Solution: The Chords' Dimensions Revealed

Therefore, the lengths of the two mutually perpendicular chords are 22 cm and 12 cm.

Conclusion: A Geometric Triumph

In this article, we embarked on a geometric quest, unraveling the lengths of two mutually perpendicular chords within a circle. By employing the Pythagorean Theorem, the Perpendicular Bisector Theorem, and strategic problem-solving techniques, we successfully navigated the intricacies of the problem. The lengths of the chords, 22 cm and 12 cm, stand as a testament to the power of geometric reasoning and the beauty of mathematical solutions. This problem serves as a reminder that within the realm of geometry, hidden relationships and elegant solutions await those who dare to explore.