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Chapter 13: Problem 6

Fill in the blanks. A________________ is a line that intersects a circle at one point.

Short Answer

Expert verified
The blank should be filled with "tangent."

Step by step solution

01

Understand the Definition

A line that intersects a circle at exactly one point is a special type of line in relation to the circle. It does not cut across the circle or pass through another point on the circle.
02

Recall the Term

The term used to describe a line that meets a circle at only one point is called a "tangent." A tangent to a circle touches the circle at exactly one point, never entering the circle's interior.

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Key Concepts

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

Tangent
A tangent is a unique concept in geometry, describing a specific kind of relationship between a line and a circle. If a line touches a circle at exactly one point and does not cross through the interior of the circle, it's called a tangent line. This point where the line makes contact with the circle is known as the point of tangency.

  • Important: The tangent is always perpendicular to the radius drawn to the point of tangency, forming a right angle (90 degrees).
  • Visualization: Imagine gently putting a pencil tip to a rubber band stretched in a circular shape鈥攊t only touches at one precise spot.
  • Applications: Tangents are used in various fields like engineering and architecture to solve problems involving circles and curves.
Understanding tangents helps in grasping other complex geometric concepts, making it a foundational element in the study of circles.
Circle
A circle is a simple yet profound shape in geometry, defined as the set of all points in a plane that are equidistant from a central point known as the center. This fixed distance from the center to any point on the circle is called the radius.

  • Key terms: The diameter is twice the length of the radius and passes through the center, dividing the circle into two equal halves.
  • Equations: A circle's equation in a Cartesian coordinate plane can be given by \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.
  • Properties: All radii of a circle are equal, and the circumference is the distance around it, calculated as \( 2\pi r \).
Circles are everywhere: they're fundamental not only in mathematics but also in nature and technology.
Line
A line in geometry is a straight one-dimensional figure that extends infinitely in both directions. Unlike segments or rays, lines have no beginning or end and are an abstract concept used to define points, angles, and shapes.

  • Properties: Lines are identified with some fundamental characterizations of slope and intercepts in coordinate geometry.
  • Equation Form: A typical linear equation can be expressed as \( y = mx + b \), where \(m\) represents the slope and \(b\) indicates the y-intercept.
  • Role in Geometry: Lines can intersect, be parallel, or be perpendicular, forming the backbone for more complex shapes.
Understanding lines is crucial not only in geometry but also in real-world applications such as navigation, construction, and computer graphics.

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Most popular questions from this chapter

Solve each system of equations by elimination for real values of x and y. $$ \left\\{\begin{array}{l} 2 x^{2}+y^{2}=6 \\ x^{2}-y^{2}=3 \end{array}\right. $$

Write the equation of a circle with a diameter whose endpoints are at \((-5,4)\) and \((7,-3)\)

Solve the system \(\left\\{\begin{array}{l}x^{2}-y^{2}=16 \\\ x^{2}+y^{2}=9\end{array}\right.\) over the complex numbers.

Write each equation in standard form, if it is not alreacty so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ x=4 y^{2} $$

Use a nonlinear system of equations to solve each problem. Number Problem. The sum of the squares of two numbers is \(221,\) and the sum of the numbers is 9. Find the numbers.
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