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Chapter 10: Problem 12

Write an equation for each circle described below. $$\text { center at }(0,0), d=12$$

Short Answer

Expert verified
The equation of the circle is \(x^2 + y^2 = 36\).

Step by step solution

01

Understanding the Equation of a Circle

The standard equation of a circle with center \((h, k)\) and radius \(r\) is given by \((x - h)^2 + (y - k)^2 = r^2\). In this exercise, the center of the circle \((h, k)\) is \((0,0)\).
02

Calculating the Radius

The problem states the diameter \(d = 12\). The radius \(r\) is half of the diameter. Therefore, \(r = \frac{12}{2} = 6\).
03

Substituting into the Circle Equation Formula

Substitute \(h=0\), \(k=0\), and \(r=6\) into the standard circle equation. The equation of the circle becomes: \[(x - 0)^2 + (y - 0)^2 = 6^2\] Simplifying this gives: \(x^2 + y^2 = 36\).

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Key Concepts

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

Understanding Circle Equations
In geometry, the equation of a circle is a fundamental concept used to describe all the points that make up a circle in a coordinate plane. The standard form of this equation is \((x - h)^2 + (y - k)^2 = r^2\), where:
  • \(h\) and \(k\) are the coordinates of the center of the circle.
  • \(r\) is the radius of the circle.
This equation ensures that every point \((x, y)\) on the circle is exactly \(r\) units away from the center \((h, k)\). The terms \((x - h)^2\) and \((y - k)^2\) represent the squared horizontal and vertical distances from the center of the circle, respectively. When these are summed and equated to \(r^2\), they describe the very essence of a circle鈥攃onstant distance from a single point, the center.
Steps for Radius Calculation
Calculating the radius from given information about a circle is often straightforward but essential. The radius \(r\) is half the diameter \(d\), which can be derived from the formula:
  • \(r = \frac{d}{2}\)
In the original exercise, the diameter of the circle is provided as 12. Therefore, the calculation to find the radius is as simple as dividing the diameter by two:
  • \(r = \frac{12}{2} = 6\)
Knowing the radius is vital when substituting into the circle's equation, ensuring one accurately represents the circle's size in the coordinate plane. This basic calculation highlights an important skill needed in solving numerous geometry problems involving circles.
Approaching Geometry Problems with Circles
Geometry problems that involve circles often require a balanced understanding of analytical and graphical approaches. Solving these problems successfully involves:
  • Identifying the center of the circle, which can be seen directly from the equation format \((h, k)\).
  • Calculating or using the given radius \(r\).
  • Applying the circle equation \((x - h)^2 + (y - k)^2 = r^2\) to determine relationships between points, chords, tangents, and other geometric entities related to the circle.
The final form is usually simplified to make it more usable for further calculations. In our exercise, once appropriate substitutions were made, the equation simplified to \(x^2 + y^2 = 36\), indicating the radius squared as 36. Recognizing the steps and logic behind these operations enables deeper insights into both simple and complex geometry problems.

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Most popular questions from this chapter

Organic molecules have geometric shapes. How many different lines of reflectional symmetry does this benzene molecule have? Does it have rotational symmetry? Sketch your answers.

REASONING Explain how the definition of a circle leads to its equation.

REVIEW In the system of equations \(4 x+3 y=6\) and \(-5 x+2 y=13,\) which expression can be substituted for \(x\) in the equation \(4 x+3 y=6 ?\) F \(\frac{3}{2}-\frac{3}{4} y\) G \(6-3 y\) \mathbf{H}-\frac{13}{5}+\frac{2}{5} y (J) 13-2 y

PROOF Write a two-column proof. Given: \(\overline{R Q}\) bisects \(\angle S R T\) Prove: \(m \angle S Q R>m \angle S R Q\)

REVIEW Larry's Fish Food comes in tubes that have a radius of 2 centimeters and a height of 7 centimeters. Odell bought a full tube, but he thinks he's used about \(\frac{1}{4}\) of it. About how much is left? F \(88 \mathrm{cm}^{3}\) G \(66 \mathrm{cm}^{3}\) H \(53 \mathrm{cm}^{3}\) I \(41 \mathrm{cm}^{3}\)
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