Find the length of the radius

Oct 13, 2025 by windailys.com 15 views

*159 Q.5. O is the centre of the circle. Find the length of radius if the chord of length 24 cm is at a what distance of 9 cm from the centre of the circle.

As an 8th standard student, you are likely familiar with basic geometry concepts, including circles and chords. In this problem, we are given a circle with center O and a chord of length 24 cm. The chord is at a distance of 9 cm from the center of the circle. Our goal is to find the length of the radius of the circle.

Visualizing the Problem

To solve this problem, let's first visualize the scenario. We have a circle with center O and a chord AB of length 24 cm. The chord is at a distance of 9 cm from the center of the circle. We can draw a perpendicular line from the center O to the chord AB, which intersects the chord at point M. This line OM is the distance from the center of the circle to the chord.

Key Concepts

Before we dive into the solution, let's review some key concepts related to circles and chords.

A chord is a line segment that connects two points on a circle.
The perpendicular bisector of a chord passes through the center of the circle.
The distance from the center of the circle to a chord is called the perpendicular distance.

Solution

Now that we have a clear understanding of the problem and the key concepts, let's move on to the solution.

Draw a diagram: Draw a diagram of the circle with center O and the chord AB of length 24 cm. Draw a perpendicular line from the center O to the chord AB, which intersects the chord at point M.
Find the length of OM: Since the chord AB is at a distance of 9 cm from the center of the circle, the length of OM is 9 cm.
Find the length of AM: Since the chord AB is a line segment, we can use the Pythagorean theorem to find the length of AM. Let's call the length of AM as x. We can write the equation as:

x^2 + 9^2 = 12^2

Simplifying the equation, we get:

x^2 = 144 - 81

x^2 = 63

x = √63

x ≈ 7.94 cm
Find the length of the radius: Now that we have the length of AM, we can find the length of the radius. The radius is the distance from the center of the circle to the point A. We can use the Pythagorean theorem again to find the length of the radius. Let's call the length of the radius as r. We can write the equation as:

r^2 = 12^2 + (√63)^2

Simplifying the equation, we get:

r^2 = 144 + 63

r^2 = 207

r = √207

r ≈ 14.37 cm

Conclusion

In this problem, we were given a circle with center O and a chord of length 24 cm. The chord was at a distance of 9 cm from the center of the circle. We were asked to find the length of the radius of the circle. Using the Pythagorean theorem, we were able to find the length of the radius, which is approximately 14.37 cm.

Key Takeaways

The length of the radius of a circle can be found using the Pythagorean theorem.
The distance from the center of the circle to a chord is called the perpendicular distance.
The perpendicular bisector of a chord passes through the center of the circle.

Practice Problems

If you want to practice more problems like this, here are a few suggestions:

Find the length of the radius of a circle with center O and a chord of length 30 cm. The chord is at a distance of 12 cm from the center of the circle.
Find the length of the radius of a circle with center O and a chord of length 20 cm. The chord is at a distance of 8 cm from the center of the circle.

Real-World Applications

Understanding the concept of circles and chords has many real-world applications. Here are a few examples:

Architecture: Architects use the concept of circles and chords to design buildings and bridges.
Engineering: Engineers use the concept of circles and chords to design machines and mechanisms.
Art: Artists use the concept of circles and chords to create symmetrical and balanced designs.

Conclusion

Q1: What is a chord in a circle?

A chord is a line segment that connects two points on a circle. It is a straight line that passes through two points on the circumference of the circle.

Q2: What is the perpendicular bisector of a chord?

The perpendicular bisector of a chord is a line that passes through the center of the circle and is perpendicular to the chord. It divides the chord into two equal parts.

Q3: How do I find the length of the radius of a circle with a given chord and distance from the center?

To find the length of the radius of a circle with a given chord and distance from the center, you can use the Pythagorean theorem. Let's say the chord is of length 24 cm and the distance from the center is 9 cm. You can draw a diagram and use the Pythagorean theorem to find the length of the radius.

Q4: What is the formula for finding the length of the radius of a circle with a given chord and distance from the center?

The formula for finding the length of the radius of a circle with a given chord and distance from the center is:

r^2 = c^2 + d^2

where r is the length of the radius, c is the length of the chord, and d is the distance from the center.

Q5: How do I use the Pythagorean theorem to find the length of the radius of a circle with a given chord and distance from the center?

To use the Pythagorean theorem to find the length of the radius of a circle with a given chord and distance from the center, you can follow these steps:

Draw a diagram of the circle with the given chord and distance from the center.
Use the Pythagorean theorem to find the length of the radius.
Simplify the equation to find the length of the radius.

Q6: What are some real-world applications of understanding circles and chords?

Understanding circles and chords has many real-world applications, including:

Architecture: Architects use the concept of circles and chords to design buildings and bridges.
Engineering: Engineers use the concept of circles and chords to design machines and mechanisms.
Art: Artists use the concept of circles and chords to create symmetrical and balanced designs.

Q7: How can I practice problems like this to improve my understanding of circles and chords?

To practice problems like this, you can try the following:

Find the length of the radius of a circle with a given chord and distance from the center.
Use the Pythagorean theorem to find the length of the radius.
Simplify the equation to find the length of the radius.

Q8: What are some common mistakes to avoid when finding the length of the radius of a circle with a given chord and distance from the center?

Some common mistakes to avoid when finding the length of the radius of a circle with a given chord and distance from the center include:

Not using the Pythagorean theorem correctly.
Not simplifying the equation correctly.
Not drawing a diagram to visualize the problem.

Q9: How can I use technology to help me understand circles and chords?

You can use technology to help you understand circles and chords by:

Using online calculators to find the length of the radius.
Using geometry software to visualize the problem.
Using online resources to learn more about circles and chords.

Q10: What are some advanced topics related to circles and chords that I can explore?

Some advanced topics related to circles and chords that you can explore include:

Circle theorems: These are theorems that relate to the properties of circles, such as the inscribed angle theorem and the central angle theorem.
Chord properties: These are properties of chords, such as the fact that the perpendicular bisector of a chord passes through the center of the circle.
Circle geometry: This is the study of the properties and relationships of circles and their parts, such as the radius, diameter, and circumference.