Parts of the Circle: True / false

A radius is a straight line that connects the center of the circle with a point on the circle itself.

Therefore, the answer is incorrect, as there are infinite radii.

To determine if the statement "If the radius of a circle is 5 cm, then the length of the diameter is 10 cm" is true, we need to use the relationship between the radius and diameter of a circle.

The diameter $d$ of a circle is calculated using the formula:

$d = 2r$

where $r$ is the radius. In this problem, the radius $r$ is given as 5 cm.

Using the formula, the diameter is:

$d = 2 \times 5 \, \text{cm} = 10 \, \text{cm}$

This matches exactly the length of the diameter given in the problem.

Therefore, the statement "If the radius of a circle is 5 cm, then the length of the diameter is 10 cm" is True.

To determine the truth of the statement, we must consider the precise definition of a chord in the context of circle geometry:

A chord is specifically defined as a line segment whose endpoints both lie on a circle. This segment connects two distinct points on the circumference of the circle. This definition highlights the role of the chord as a geometric entity within the circle.

Given this definition, we evaluate the statement: "A chord is a segment that connects two points on a circle."

The provided statement accurately describes the nature of a chord. The endpoints of the segment must be on the circle, thus aligning perfectly with the standard definition of a chord.

Therefore, the statement is True.

The diameter of a circle is defined as the distance across the circle through its center. It is directly related to the radius, which is the distance from the center to a point on the circumference of the circle.

By the standard definition in geometry, the diameter ($d$) of a circle is expressed in terms of its radius ($r$) as:

$d = 2r$

This equation clearly states that the diameter is twice the length of the radius. Hence, the statement, “The diameter of a circle is twice as long as its radius,” aligns with this fundamental geometric property.

Therefore, the statement is True.

To solve the problem, we will explore the properties of diameters and circles:

• Step 1: Define a diameter - A diameter is a line segment that passes through the center of the circle and has its endpoints on the circle.
• Step 2: Consider the properties of a circle - A circle is perfectly symmetric around its center.
• Step 3: Analyze rotational possibilities - Due to its symmetry, a circle can be rotated around its center any number of times, and each rotation aligns a potential diameter with another.

Now, let's examine these points step-by-step:
Step 1: A diameter requires only that a line passes through the center of the circle and touches both sides of the circle.
Step 2: Because of rotational symmetry, once we have one diameter, we can rotate it by any arbitrary angle $\theta$ (where $0 \leq \theta < 360$ degrees), and it still qualifies as a diameter.
Step 3: Since $\theta$ can take infinitely many values between $0$ and $360$ degrees (conceptually covering a continuum of angles), a circle can indeed have infinitely many diameters.

Therefore, the statement that a circle has infinite diameters is \textbf{True}. This leads us to the conclusion that the correct choice is Choice 1: True.

To solve this problem, we first review the standard definition of a circle's diameter. By definition, a diameter of a circle is a straight line segment that passes through the center of the circle and has its endpoints on the circle itself.

Let's compare this with the given statement:
- The statement says the diameter connects two points on the circle. This aligns with the standard definition.

- The statement says the diameter passes through the center of the circle. This also aligns with the standard definition.

Therefore, the statement correctly describes the properties of a diameter.

Consequently, the statement is True.

To solve this question, we must understand the definitions of the terms "radius" and "chord" in the context of a circle:

• A radius is a line segment that connects the center of the circle to any point on the circle's circumference. All radii of a circle are equal in length.
• A chord is a line segment whose endpoints both lie on the circle's circumference. The chord does not necessarily pass through the center of the circle, and chords can have different lengths.

Given these definitions, observe the following points:

• The radius is inherently different from the general concept of a chord because the radius must include the circle's center as one of its points, while a chord only specifies that both endpoints lie on the circle's edge, offering no requirement to pass through the center.
• An important sub-case is the diameter, which is a special chord that does pass through the center and is equal to twice the radius ($2r$). However, while the diameter is indeed a chord, the radius itself cannot be viewed as such because it does not completely lie between two points on the circle but instead starts from the center.

Hence, the statement that "The radius of a circle is the chord" is false because a radius does not fulfill the general definition of a chord, which requires two endpoints on the circle's circumference that do not include the center of the circle.

Therefore, the correct choice is False.