To find the radius of the circle when the length of a tangent from a point A (5 cm from the center of the circle) is 4 cm, we can use the Pythagorean theorem.
Triangle Formation: Point A, the center of the circle (let’s call it O), and the point of tangency on the circle (let’s call it B) form a right-angled triangle AOB.
Radius and Tangent: The radius OB is perpendicular to the tangent AB at the point of tangency.
Applying Pythagorean Theorem: In triangle AOB, AO² = OB² + AB².
Substituting Values: 5² = OB² + 4² (since AO = 5 cm, AB = 4 cm).
Calculating Radius: 25 = OB² + 16 ⇒ OB² = 25 − 16 ⇒ OB² = 9.
Radius: OB = √9 = 3 cm.
Therefore, the radius of the circle is 3 cm.
Let’s discuss in detail
Introduction to the Problem
In the study of circle geometry, one often encounters problems that involve finding the radius of a circle given certain conditions. A common type of problem is determining the radius when the length of a tangent from an external point and the distance of this point from the circle’s center are known. This problem not only tests one’s understanding of geometric principles but also their ability to apply the Pythagorean theorem in a practical context.
Understanding the Given Data
In the problem presented, we have two key pieces of information: the length of the tangent from a point A to the circle is 4 cm, and the distance from A to the center of the circle is 5 cm. These two data points are crucial as they form two sides of a right-angled triangle. The point where the tangent touches the circle forms a right angle with the radius at that point. This right angle is fundamental to applying the Pythagorean theorem, which is central to solving this problem.
The Role of the Pythagorean Theorem
The Pythagorean theorem is a cornerstone of geometry, stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In our scenario, the hypotenuse is the line segment from point A to the center of the circle, and the other two sides are the radius of the circle and the length of the tangent from A to the circle.
Calculating the Radius
To find the radius of the circle, we set up an equation based on the Pythagorean theorem. The square of the distance from A to the center (5 cm) equals the sum of the squares of the radius (unknown) and the length of the tangent (4 cm). Mathematically, this is represented as 5² = r² + 4². Solving this equation will give us the value of the radius.
Solving the Equation
Substituting the known values into the equation, we get 25 = r² + 16. Rearranging the equation to solve for r², we find r² = 25 − 16, which simplifies to r² = 9. The final step is to find the square root of 9, which yields r = 3. This calculation is straightforward but requires careful attention to ensure accuracy.
The radius of the circle, in this case, is found to be 3 cm. This problem is a classic example of applying the Pythagorean theorem in a geometric context. It demonstrates how a seemingly complex problem can be broken down into simpler parts using fundamental principles of mathematics. Such problems not only enhance one’s problem-solving skills but also deepen their understanding of how geometry is applied in various scenarios.