Calculating the Length of the Chord

Oct 13, 2025 by miodeya.com 92 views

In this problem, we are dealing with two concentric circles, which means they share the same center. The larger circle has a radius of 5 cm, while the smaller circle has a radius of 3 cm. Our goal is to find the length of the chord of the larger circle that touches the smaller circle.

Visualizing the Problem

To approach this problem, let's first visualize the scenario. We have two concentric circles, with the larger circle having a radius of 5 cm and the smaller circle having a radius of 3 cm. The chord of the larger circle that touches the smaller circle is the line segment that connects two points on the circumference of the larger circle and passes through the center of the smaller circle.

Key Concepts: Tangents and Chords

Before we dive into the solution, let's review some key concepts related to circles. A tangent to a circle is a line that touches the circle at a single point, called the point of tangency. A chord of a circle is a line segment that connects two points on the circumference of the circle.

In this problem, we are dealing with a chord that touches the smaller circle. This means that the chord is tangent to the smaller circle. We can use this information to find the length of the chord.

Solution: Using the Pythagorean Theorem

To find the length of the chord, we can use the Pythagorean theorem. Let's draw a diagram to represent the situation.

  A
  / \
 /   \
/_____\
|     |
|  O  |
|     |
 \_____/
  B

In this diagram, O is the center of the smaller circle, and AB is the chord of the larger circle that touches the smaller circle. We can draw a radius from the center of the smaller circle to the point of tangency, which we'll call P.

  A
  / \
 /   \
/_____\
|     |
|  O  |
|  |  |
|  P  |
|     |
 \_____/
  B

Now, we can use the Pythagorean theorem to find the length of the chord. The Pythagorean theorem states 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 lengths of the other two sides.

In this case, we have a right-angled triangle with the radius of the smaller circle (3 cm) as one side, the radius of the larger circle (5 cm) as the other side, and the chord (AB) as the hypotenuse. We can use the Pythagorean theorem to find the length of the chord.

Calculating the Length of the Chord

Using the Pythagorean theorem, we can calculate the length of the chord as follows:

AB^2 = OP^2 + PB^2
AB^2 = 3^2 + (5^2 - 3^2)
AB^2 = 9 + 16
AB^2 = 25
AB = sqrt(25)
AB = 5

Therefore, the length of the chord of the larger circle that touches the smaller circle is 5 cm.

Conclusion

In this problem, we used the Pythagorean theorem to find the length of the chord of the larger circle that touches the smaller circle. We visualized the problem, reviewed key concepts related to circles, and used the Pythagorean theorem to calculate the length of the chord. The length of the chord is 5 cm.

Key Takeaways

Two concentric circles are circles that share the same center.
A tangent to a circle is a line that touches the circle at a single point.
A chord of a circle is a line segment that connects two points on the circumference of the circle.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Practice Problems

Two concentric circles have radii of 4 cm and 2 cm. Find the length of the chord of the larger circle that touches the smaller circle.
A chord of a circle has a length of 6 cm. If the radius of the circle is 4 cm, find the distance from the center of the circle to the chord.
Two concentric circles have radii of 6 cm and 3 cm. Find the length of the chord of the larger circle that touches the smaller circle.

References

"Geometry" by Michael Artin
"Calculus" by Michael Spivak
"Mathematics for Computer Science" by Eric Lehman and Tom Leighton
Q&A: Two Concentric Circles and the Length of a Chord =====================================================

In the previous article, we discussed how to find the length of the chord of the larger circle that touches the smaller circle. In this article, we will answer some frequently asked questions related to the problem.

Q: What are concentric circles?

A: Concentric circles are circles that share the same center. In other words, they are circles that have the same center point and different radii.

Q: What is a tangent to a circle?

A: A tangent to a circle is a line that touches the circle at a single point, called the point of tangency. In the context of the problem, the chord of the larger circle that touches the smaller circle is a tangent to the smaller circle.

Q: What is a chord of a circle?

A: A chord of a circle is a line segment that connects two points on the circumference of the circle. In the context of the problem, the chord of the larger circle that touches the smaller circle is a chord of the larger circle.

Q: How do I find the length of the chord of the larger circle that touches the smaller circle?

A: To find the length of the chord of the larger circle that touches the smaller circle, you can use the Pythagorean theorem. The Pythagorean theorem states 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 lengths of the other two sides.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical formula that describes the relationship between the lengths of the sides of a right-angled triangle. It states 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 lengths of the other two sides.

Q: How do I apply the Pythagorean theorem to find the length of the chord?

A: To apply the Pythagorean theorem to find the length of the chord, you need to draw a diagram that represents the situation. In this case, you can draw a right-angled triangle with the radius of the smaller circle as one side, the radius of the larger circle as the other side, and the chord as the hypotenuse. Then, you can use the Pythagorean theorem to find the length of the chord.

Q: What is the length of the chord of the larger circle that touches the smaller circle?

A: The length of the chord of the larger circle that touches the smaller circle is 5 cm.

Q: Can I use the Pythagorean theorem to find the length of the chord in other situations?

A: Yes, you can use the Pythagorean theorem to find the length of the chord in other situations. The Pythagorean theorem is a general formula that can be applied to any right-angled triangle.

Q: What are some other applications of the Pythagorean theorem?

A: The Pythagorean theorem has many applications in mathematics and science. Some examples include:

Finding the length of the hypotenuse of a right-angled triangle
Finding the distance between two points in a coordinate plane
Finding the height of a building or a mountain
Finding the length of a shadow or a reflection

Conclusion

In this article, we answered some frequently asked questions related to the problem of finding the length of the chord of the larger circle that touches the smaller circle. We discussed the concept of concentric circles, tangents, and chords, and we applied the Pythagorean theorem to find the length of the chord. We also discussed some other applications of the Pythagorean theorem.

Key Takeaways

Concentric circles are circles that share the same center.
A tangent to a circle is a line that touches the circle at a single point.
A chord of a circle is a line segment that connects two points on the circumference of the circle.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
The Pythagorean theorem has many applications in mathematics and science.

Practice Problems

Two concentric circles have radii of 4 cm and 2 cm. Find the length of the chord of the larger circle that touches the smaller circle.
A chord of a circle has a length of 6 cm. If the radius of the circle is 4 cm, find the distance from the center of the circle to the chord.
Two concentric circles have radii of 6 cm and 3 cm. Find the length of the chord of the larger circle that touches the smaller circle.

References

"Geometry" by Michael Artin
"Calculus" by Michael Spivak
"Mathematics for Computer Science" by Eric Lehman and Tom Leighton