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Chapter 10: Problem 12

Find the radius of a circle if a \(24-\mathrm{cm}\) chord is \(9 \mathrm{cm}\) from the center.

Short Answer

Expert verified
The radius of the circle is 15 cm

Step by step solution

01

Understand the problem

A chord of length 24 cm is given, from which a line has been drawn 9 cm from the center of the circle. This line perpendicular to the chord from the center bisects the chord and forms a right triangle in the circle, where the hypotenuse is the radius (r), the perpendicular from the center to the chord is one of the sides (9 cm) and half of the chord is the other side (24/2 = 12 cm)
02

Apply the Pythagorean theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as: \(a^2 + b^2 = c^2\), where c represents the length of the hypotenuse. In this case, \(r^2 = 9^2 + 12^2\)
03

Solve for the Radius

By solving the above equation we find that \(r^2 = 81 + 144 = 225\). Taking the square root of both sides gives \(r = \sqrt{225} = 15\) cm

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chord Length
A chord is a straight line connecting two points on the circumference of a circle. In our scenario, the chord length is given as 24 cm. To visualize this, imagine a line cutting across a circle and touching its boundary at both ends. This line is known as a chord. Chords are significant in circle geometry as they create segments within the circle.

When a perpendicular line is drawn from the center of the circle to the chord, it bisects the chord, meaning it splits the chord into two equal parts. So, in this exercise, if we draw a perpendicular from the center to the 24 cm chord, it divides it into two equal segments of 12 cm each.

Understanding chords is crucial because they help us apply various geometric principles, such as the Pythagorean Theorem, to solve problems associated with circles.
Pythagorean Theorem
The Pythagorean Theorem is pivotal in many areas of geometry, especially when dealing with right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides.

The Pythagorean Theorem is usually expressed as:
  • \[ a^2 + b^2 = c^2 \]
where \( c \) is the hypotenuse, and \( a \) and \( b \) represent the triangle's legs.

In our exercise, the radius of the circle is the hypotenuse, and the two legs are composed of 9 cm, which represents the perpendicular distance from the center of the circle to the chord, and 12 cm, which is half of the chord length. Applying the theorem helps us calculate the hypotenuse, which is the circle's radius in this context.
Right Triangle
A right triangle is a triangle where one of its angles is exactly 90 degrees. This property makes right triangles special because it allows us to apply the Pythagorean Theorem to solve for unknown lengths.

In the circle geometry problem, the right triangle is formed where:
  • One leg is the distance from the center of the circle (9 cm).
  • The other leg is half of the chord (12 cm).
  • The hypotenuse, which is what we want to find, is the radius of the circle.
The right angle occurs at the intersection point where the 9 cm line meets the chord perpendicularly.

Understanding the configuration and properties of right triangles is essential when working with circles, as they frequently arise in problems involving chords and radii.

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Most popular questions from this chapter

Find the length of each arc described. (The length is a fractional part of the circumference.) a An arc that is \(\frac{5}{8}\) of the circumference of a circle with radius 12 b An arc that has a measure of 270 and is part of a circle with radius 12

A secant-secant angle intercepts arcs that are \(\frac{3}{5}\) and \(\frac{3}{8}\) of the circle. If a chord-chord angle and its vertical angle intercept the same arcs, what is the measure of the chord-chord angle?

A circle is inscribed in a triangle with sides \(8,10,\) and \(12 .\) The point of tangency of the 8-unit side divides that side in the ratio \(x: y,\) where \(x

Given: \(\odot F, \overline{A B} \cong \overline{A C}\) \(\overline{\mathrm{DF}} \perp \overline{\mathrm{AB}}, \overline{\mathrm{EF}} \perp \overline{\mathrm{AC}}\) Prove: \(\triangle \mathrm{ADE}\) is isosceles.

Discuss the location of the center of a circle circumscribed about each of the following types of triangles. a Right b Acute c Obtuse
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