/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Write the equation of a circle t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 19

Write the equation of a circle that has a diameter of 9 units and its center at \((6,-21)\).

Short Answer

Expert verified
The equation of the circle is \((x - 6)^2 + (y + 21)^2 = 20.25\).

Step by step solution

01

Find the Radius

To find the radius of a circle when you know the diameter, use the formula: \( r = \frac{d}{2} \). Here, the diameter \( d \) is given as 9 units. So, \( r = \frac{9}{2} = 4.5 \).
02

Use the Circle Equation Formula

The standard equation of a circle with center at \((h, k)\) and radius \( r \) is \((x - h)^2 + (y - k)^2 = r^2\). Substituting in the center \((h, k) = (6, -21)\) and \( r = 4.5 \), the equation becomes: \((x - 6)^2 + (y + 21)^2 = (4.5)^2 \).
03

Simplify the Equation

Calculate \((4.5)^2\) to simplify the equation. \((4.5)^2 = 20.25\). Substitute this back into the circle equation to get the final form: \((x - 6)^2 + (y + 21)^2 = 20.25\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Geometry Concepts
Geometry revolves around understanding shapes, sizes, and the properties of space. Circles, in particular, are a fundamental part of geometry that focuses on round shapes with all points on the boundary equidistant from a central point. This property of circles leads to unique equations that define their geometry.
  • A circle can be fully defined by its center, which is the point equidistant from all points on the circle, and its radius, the distance from the center to any point on the circle.
  • The diameter, which is twice the radius, spans the circle through its center, connecting two points on its boundary.
  • The equation of a circle encapsulates these parameters, allowing us to describe its size and position in a coordinate plane.
Geometry concepts are crucial as they provide the framework for solving problems relating to shapes and sizes, such as finding the equation for a circle with a specified center and diameter.
Radius and Diameter
Understanding the radius and diameter is key to mastering circle-related problems. These two related measurements can often be confused but have distinct roles in defining a circle.
  • The radius (\( r \)) is the distance from the center of the circle to its edge. It is half the diameter.
  • The diameter (\( d \)) is the longest distance across the circle, passing through the center. It is twice the radius.
  • To find the radius when the diameter is known, use the formula \( r = \frac{d}{2} \), which halves the diameter to get the radius.
In our exercise, given a diameter of 9 units, the radius is calculated as \( r = \frac{9}{2} = 4.5 \) units. With these measurements, we can write the standard equation of a circle, crucial in many geometry tasks.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, combines algebra and geometry using coordinate planes to describe geometric figures and their properties. For circles, this involves using the circle's center and radius in a coordinate system.
  • The standard equation of a circle in coordinate geometry is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
  • This equation helps us determine if a point lies on the circle by substituting the point鈥檚 coordinates into the equation.
  • In our original problem, we were given a center \((6, -21)\) and a radius of 4.5 units. Substituting these values into the equation gives \((x - 6)^2 + (y + 21)^2 = 20.25\), fully describing the circle鈥檚 position and size.
Coordinate geometry thus provides a powerful tool for working with circles in a two-dimensional plane, enabling us to clearly express the circle's equation and analyze its properties.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use substitution to solve each system of equations. $$ \begin{aligned} &y=x-1 \\ &3 x-6 y=16 \end{aligned} $$

Solve each system of equations by graphing. $$ \begin{aligned} &y=\frac{1}{2} x-4 \\ &y=-x+5 \end{aligned} $$

Art A frieze is a pattern running across the upper part of a wall. Make a frieze by drawing a basic pattern on grid paper and then translating it as many times as you can.

Find each product. $$(10,-6.5) \times 0.5$$

Find the value of \(y\) for each given value of \(x\). $$ \begin{aligned} &9 y=11-x \\ &x=y+3 \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.