Exploring the Geometry of Concentric Circles: Tangent Chords and Pythagorean Theorem

When dealing with geometric shapes, particularly when circles are involved, understanding their properties and how they interact is crucial. One interesting scenario is when two circles share the same center, known as concentric circles. In this article, we will delve into a specific problem involving the geometry of concentric circles, focusing on a tangent chord and the application of the Pythagorean theorem.

The Geometry of Concentric Circles and Tangent Chords

Two concentric circles are drawn with radii of 12 cm and 13 cm. The larger circle shares the same center as the smaller circle. A chord of the larger circle is tangent to the smaller circle, and we aim to find the length of this chord. Understanding the properties of such geometric configurations is essential for solving a variety of problems in mathematics and geometry.

Geometric Configuration and Pythagorean Theorem

To solve the problem, we must first understand the geometric configuration of the circles. Let's denote the center of the circles as O.

Consider the chord AB of the larger circle, which is tangent to the smaller circle at point P. Since P is the point of tangency, OP is perpendicular to AB. P is also the midpoint of AB.

Using the Pythagorean Theorem

Now, let's apply the Pythagorean theorem in the right triangle ODA or ODB. Here, D is the midpoint of AB, and OD is the radius of the smaller circle, which is 12 cm.

The length of AD represents half the length of the chord AB. According to the Pythagorean theorem:

AD2 OP2 OA2

Substituting the known values:

AD2 122 132

AD2 144 169

AD2 169 - 144

AD2 25

AD √25

AD 5 cm

Since AD is half the length of the chord AB, the full length of the chord AB is:

AB 2 × AD 2 × 5 10 cm

Conclusion

By applying the Pythagorean theorem in the right triangle ODA, we have determined that the length of any chord of the larger concentric circle that is tangent to the smaller concentration is 10 cm. This solution highlights the importance of using geometric properties and theorems to solve complex problems.

Keywords

concentric circles, tangent chord, Pythagorean theorem