Points { P, Q$}$, And { R$}$ Are Points On The Circumference Of A Circle. If { PQ = PR = 13 , \text{cm}$}$ And { QR = 10 , \text{cm}$}$, What Is The Radius Of The Circle?

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Introduction

In geometry, a circle is a set of points that are all equidistant from a central point called the center. The distance from the center to any point on the circle is called the radius. In this article, we will explore how to find the radius of a circle given the lengths of three points on its circumference.

The Problem

We are given three points, P, Q, and R, on the circumference of a circle. The lengths of the line segments PQ and PR are both 13 cm, and the length of the line segment QR is 10 cm. Our goal is to find the radius of the circle.

Understanding the Geometry

To solve this problem, we need to understand the geometry of the circle and the relationships between the points P, Q, and R. Since PQ and PR are both radii of the circle, they are equal in length. This means that the triangle PQR is an isosceles triangle, with PQ and PR being the two equal sides.

Drawing the Diagram

To better understand the problem, let's draw a diagram of the circle and the points P, Q, and R.

  +---------------+
  |               |
  |  P  (13 cm)  |
  |               |
  +---------------+
           |
           |
           v
  +---------------+
  |               |
  |  Q  (10 cm)  |
  |               |
  +---------------+
           |
           |
           v
  +---------------+
  |               |
  |  R  (13 cm)  |
  |               |
  +---------------+

Using the Pythagorean Theorem

Since we know the lengths of the sides PQ and PR, we can use the Pythagorean theorem to find the length of the side QR. 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 can use the Pythagorean theorem to find the length of the radius of the circle. Let's call the radius of the circle "r". Then, we can use the Pythagorean theorem to write:

r^2 = (13 cm)^2 - (10 cm)^2

Simplifying the Equation

Now, let's simplify the equation by evaluating the squares:

r^2 = 169 cm^2 - 100 cm^2

r^2 = 69 cm^2

Taking the Square Root

To find the value of r, we need to take the square root of both sides of the equation:

r = √(69 cm^2)

r = √(9 cm^2 * 7 cm^2)

r = 3√7 cm

Conclusion

In this article, we used the Pythagorean theorem to find the radius of a circle given the lengths of three points on its circumference. We first drew a diagram of the circle and the points P, Q, and R, and then used the Pythagorean theorem to write an equation for the radius of the circle. We simplified the equation and took the square root of both sides to find the value of the radius.

The Final Answer

The radius of the circle is 3√7 cm.

Additional Information

  • The Pythagorean theorem is a fundamental concept in geometry that can be used to solve a wide range of problems involving right-angled triangles.
  • The radius of a circle is the distance from the center of the circle to any point on its circumference.
  • The length of the radius of a circle can be found using the Pythagorean theorem, given the lengths of three points on its circumference.
    Solving a Circle Problem: Finding the Radius - Q&A =====================================================

Introduction

In our previous article, we explored how to find the radius of a circle given the lengths of three points on its circumference. We used the Pythagorean theorem to write an equation for the radius of the circle and simplified it to find the value of the radius. In this article, we will answer some frequently asked questions about the problem and provide additional information to help you better understand the concept.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that 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 use the Pythagorean theorem to find the radius of a circle?

A: To use the Pythagorean theorem to find the radius of a circle, you need to know the lengths of three points on its circumference. Let's call the points P, Q, and R. Then, you can use the Pythagorean theorem to write an equation for the radius of the circle:

r^2 = (PQ)^2 - (QR)^2

where r is the radius of the circle, PQ is the length of the line segment PQ, and QR is the length of the line segment QR.

Q: What if I don't know the length of the line segment QR?

A: If you don't know the length of the line segment QR, you can use the Pythagorean theorem to find it. Let's call the length of the line segment QR "x". Then, you can use the Pythagorean theorem to write an equation for x:

x^2 = (PQ)^2 - (PR)^2

where PQ is the length of the line segment PQ, and PR is the length of the line segment PR.

Q: Can I use the Pythagorean theorem to find the radius of a circle if the points P, Q, and R are not on the circumference of a circle?

A: No, you cannot use the Pythagorean theorem to find the radius of a circle if the points P, Q, and R are not on the circumference of a circle. The Pythagorean theorem only works for right-angled triangles, and a circle is not a right-angled triangle.

Q: What if I have a circle with a radius of 5 cm and I want to find the length of the line segment PQ?

A: To find the length of the line segment PQ, you need to know the length of the line segment QR. Let's call the length of the line segment QR "x". Then, you can use the Pythagorean theorem to write an equation for x:

x^2 = (5 cm)^2 - (PQ)^2

where PQ is the length of the line segment PQ.

Q: Can I use the Pythagorean theorem to find the length of the line segment PQ if I don't know the length of the line segment QR?

A: No, you cannot use the Pythagorean theorem to find the length of the line segment PQ if you don't know the length of the line segment QR. You need to know the length of the line segment QR to use the Pythagorean theorem to find the length of the line segment PQ.

Conclusion

In this article, we answered some frequently asked questions about the problem of finding the radius of a circle given the lengths of three points on its circumference. We provided additional information to help you better understand the concept and used the Pythagorean theorem to write equations for the radius of the circle.

The Final Answer

The radius of a circle can be found using the Pythagorean theorem, given the lengths of three points on its circumference.

Additional Information

  • The Pythagorean theorem is a fundamental concept in geometry that can be used to solve a wide range of problems involving right-angled triangles.
  • The radius of a circle is the distance from the center of the circle to any point on its circumference.
  • The length of the radius of a circle can be found using the Pythagorean theorem, given the lengths of three points on its circumference.