How To Calculate Circle Radius Given Chord Length And Distance To Center

Jul 16, 2025 by ADMIN 73 views

Calculating the Radius of a Circle Given a Chord and a Perpendicular Distance

In this article, we will delve into a classic geometry problem involving circles, chords, and radii. Specifically, we will explore how to calculate the radius of a circle when given the length of a chord and the perpendicular distance from the center of the circle to the chord. This type of problem often appears in mathematics courses and standardized tests, making it essential to understand the underlying principles and problem-solving techniques. This article provides a step-by-step solution to a specific problem, along with a detailed explanation of the geometric concepts involved. We will use the Pythagorean theorem and properties of circles to arrive at the answer, ensuring a clear and comprehensive understanding of the solution process. By the end of this article, you will be equipped with the knowledge and skills to tackle similar problems with confidence. This exploration will not only enhance your geometric intuition but also deepen your understanding of the interconnectedness of various mathematical concepts within the realm of circles and their properties.

Problem Statement

Let's consider the following problem: AC is a chord of a circle. O is the center of the circle. The length of the chord AC is 24 mm, and the distance BD from the center O to the chord AC is 6 mm. Our goal is to determine the radius of the circle. To fully justify our answer, we will need to provide a clear and logical explanation of each step in our solution, relying on established geometric principles and theorems. This problem not only requires a firm grasp of geometric concepts but also the ability to apply them methodically to arrive at the correct answer. The process of solving this problem will illuminate the relationship between the radius, chord, and the perpendicular distance from the center to the chord, reinforcing fundamental geometric principles.

Solution

To solve this problem, we will use the properties of circles and the Pythagorean theorem. Here’s a step-by-step solution:

Step 1: Draw a Diagram

Drawing a clear diagram is crucial for visualizing the problem. Start by drawing a circle with center O. Draw a chord AC across the circle. Mark a point D on the chord such that BD is perpendicular to AC. Label the lengths AC = 24 mm and BD = 6 mm. This visual representation will help us understand the relationships between the different elements of the problem and guide our solution process. A well-drawn diagram serves as a roadmap, making it easier to identify relevant geometric properties and apply the appropriate theorems. By visually representing the problem, we can break it down into smaller, more manageable parts, making the solution process more intuitive and less prone to errors. Furthermore, a diagram helps in identifying the right triangles that are essential for applying the Pythagorean theorem, a key tool in solving this problem.

Step 2: Identify Key Geometric Properties

A line drawn from the center of a circle perpendicular to a chord bisects the chord. This means that since OD is perpendicular to AC, it divides AC into two equal parts. Therefore, AD = DC. This property is fundamental to solving this problem, as it allows us to determine the length of AD, which is necessary for applying the Pythagorean theorem. Understanding this property is crucial for recognizing the symmetrical nature of the circle and how the perpendicular distance from the center interacts with the chord. It's a cornerstone concept in circle geometry and is frequently used in various problems involving chords and radii. This bisection property simplifies the problem by providing a direct relationship between the length of the chord and its segments, paving the way for further calculations.
AD = DC = AC / 2 = 24 mm / 2 = 12 mm. Now we know the length of AD, which is half the length of the chord AC. This calculation is a direct application of the bisection property mentioned above. Knowing the length of AD is a critical step, as it forms one side of a right-angled triangle that we will use to find the radius of the circle. The precise calculation ensures that we have an accurate value to work with, which is essential for the subsequent application of the Pythagorean theorem. This numerical value serves as a bridge between the geometric property and the algebraic calculations required to solve the problem.
Draw a line from the center O to point A (OA). OA is a radius of the circle, which we want to find. Connecting the center of the circle to a point on the circumference, such as A, is a standard technique in circle geometry problems. This line segment, representing the radius, forms the hypotenuse of a right-angled triangle, making it possible to use the Pythagorean theorem. Visualizing this radius as a part of the triangle is key to setting up the equation that will lead to the solution. The radius is the unknown quantity we are trying to determine, and establishing this geometric connection is a crucial step in our problem-solving strategy. By introducing the radius, we bring in the element that directly relates to the core question of the problem.

Step 3: Apply the Pythagorean Theorem

We now have a right-angled triangle ODA, where OA is the hypotenuse (the radius), AD is one side, and OD is another side. We know AD = 12 mm. To find OD, we use the information given: BD = 6 mm. Let the radius be r. Then, OD = r - BD. This step involves setting up the geometric framework for applying the Pythagorean theorem. Recognizing the right triangle ODA is crucial, as it allows us to leverage this powerful theorem to relate the sides of the triangle. The expression for OD in terms of the radius 'r' and the given distance BD is a key insight that links the known information to the unknown radius. This algebraic representation sets the stage for formulating the equation that will ultimately solve for the radius. It demonstrates the ability to translate geometric relationships into algebraic expressions, a fundamental skill in mathematical problem-solving.

Since BD = 6 mm, OD = r - 6 mm. Now we have an expression for OD in terms of the radius r. This algebraic representation is a crucial step in setting up the equation for the Pythagorean theorem. The ability to express OD as (r - 6) allows us to relate the unknown radius directly to the known distance BD. This step highlights the importance of using the given information strategically to build the necessary relationships for solving the problem. By representing OD algebraically, we can now incorporate it into the Pythagorean theorem equation, which will enable us to solve for the unknown radius.
Applying the Pythagorean theorem to triangle ODA, we get: OA² = OD² + AD². This is the core application of the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (OA) is equal to the sum of the squares of the other two sides (OD and AD). This equation forms the foundation for solving the problem. Understanding and applying the Pythagorean theorem is fundamental to solving many geometric problems, and this step demonstrates its direct relevance to this specific scenario. The correct application of the theorem is crucial for setting up the equation that will lead to the solution. It showcases the ability to translate a geometric relationship into an algebraic equation, a key skill in mathematical problem-solving.

Step 4: Substitute and Solve

Substitute the values we know into the equation: r² = (r - 6)² + 12². This step involves substituting the expressions we derived earlier for OD and AD into the Pythagorean theorem equation. By replacing the geometric quantities with their algebraic representations, we transform the problem into a solvable algebraic equation. This substitution is a critical step in bridging the gap between the geometric and algebraic aspects of the problem. It sets the stage for simplifying and solving the equation to find the unknown radius. The accuracy of the substitution is paramount to ensuring the correct solution.

Expand and simplify: r² = (r² - 12r + 36) + 144. This step involves expanding the squared term (r - 6)² and simplifying the equation. Expanding the expression correctly is crucial to avoiding algebraic errors. This simplification process aims to isolate the unknown variable 'r' and make the equation easier to solve. The algebraic manipulations performed here are essential for transforming the equation into a more manageable form. This step highlights the importance of algebraic proficiency in solving geometric problems.
r² = r² - 12r + 180. Continue simplifying by combining the constant terms. This simplification step further reduces the complexity of the equation. Combining like terms is a fundamental algebraic technique that streamlines the equation and makes it easier to solve for the unknown variable. This process ensures that the equation is in its simplest form, reducing the chances of errors in subsequent steps. The accuracy of this simplification is crucial for arriving at the correct solution.
0 = -12r + 180. Notice that r² terms cancel out, which simplifies the equation significantly. The cancellation of the r² terms is a key observation that simplifies the equation from a quadratic to a linear form. This simplification makes the equation much easier to solve for 'r'. This step demonstrates the elegance of the solution process, where algebraic manipulations lead to a more straightforward equation. Recognizing and exploiting such cancellations is a valuable skill in mathematical problem-solving.
12r = 180. Rearrange the equation to isolate the term with 'r'. This step involves isolating the term containing the unknown variable 'r' on one side of the equation. Rearranging the terms is a fundamental algebraic technique that prepares the equation for the final step of solving for 'r'. This manipulation is a necessary step in the process of finding the value of the radius. The accurate rearrangement of terms ensures that the subsequent division will yield the correct result.
r = 180 / 12 = 15 mm. Divide both sides by 12 to solve for r, which is the radius of the circle. This is the final step in solving the equation, where we divide both sides by the coefficient of 'r' to find its value. This calculation provides the numerical value of the radius, which is the solution to the problem. The accuracy of this division is crucial for arriving at the correct answer. This step demonstrates the culmination of the algebraic process, leading to the determination of the unknown radius.

Step 5: State the Answer

The radius of the circle is 15 mm. This is the final statement of the solution, clearly indicating the value of the radius that we have calculated. Stating the answer explicitly is crucial for clarity and completeness. This statement provides a definitive answer to the problem posed and leaves no ambiguity about the result. It is the culmination of the entire problem-solving process, presenting the final numerical solution in a clear and concise manner. This step ensures that the solution is easily understood and can be readily applied in other contexts.

Conclusion

In conclusion, by applying the properties of circles and the Pythagorean theorem, we have successfully calculated the radius of the circle to be 15 mm. This problem demonstrates the importance of understanding geometric principles and their application in solving real-world problems. The process involved drawing a diagram, identifying key geometric properties, applying the Pythagorean theorem, and solving the resulting equation. Each step was crucial in arriving at the correct answer. This problem not only reinforces the understanding of circle geometry but also highlights the interconnectedness of geometry and algebra in mathematical problem-solving. The ability to visualize the problem, identify relevant properties, and apply the appropriate theorems is essential for success in mathematics. By mastering these skills, one can approach more complex problems with confidence and achieve accurate solutions.