Lesson Explainer: Positions of Points, Straight Lines, and Circles with respect to Circles Mathematics

In this explainer, we will learn how to find the positions of points, straight lines, and circles with respect to other circles.

Recall that, mathematically, we define a circle as being a set of points in a plane that are a constant distance from a point in the center.

The line segment from the center to a point on the circumference is called a radius. It is common to denote the length of the radius as 𝑟.

To begin with, let us consider how points can be positioned with regard to a circle. Consider the diagram below.

There are three distinct possibilities in terms of where points can exist on a plane with respect to the circle:

  1. inside the circle (e.g., point 𝐴)
  2. on the circle (e.g., point 𝐵)
  3. outside the circle (e.g., point 𝐶)

These distinctions matter in terms of the distance that points can have from the center of the circle, compared to the radius. For example, let us consider three points on the same straight line, as shown below.

We can see that 𝑀𝐴<𝑟, while 𝑀𝐵=𝑟 and 𝑀𝐶>𝑟. We can generalize this to any points on the plane with the following rule.

Rule: Distance of Points from the Center of a Circle

For a circle with center 𝑀 and radius 𝑟, and a general point 𝑃,

  • if 𝑀𝑃<𝑟, then 𝑃 is inside the circle;
  • if 𝑀𝑃=𝑟, then 𝑃 is on the circle;
  • if 𝑀𝑃>𝑟, then 𝑃 is outside the circle.

Let us see an application of this rule in the following example.

Example 1: Using the Position of a Point with respect to a Circle to Solve an Inequality

A circle has a radius of 90 cm. A point lies on the circle at a distance of (3𝑥3) cm from the center. Which of the following is true?

  1. 𝑥<31
  2. 𝑥=31
  3. 𝑥>31

Answer

Recall that if a point lies on a circle, then its distance from the center is equal to the radius. This gives us the following linear equation: 90=3𝑥3.

Although it is not strictly necessary, let us illustrate this by drawing a diagram of the problem. Let 𝑀 be the center of the circle and 𝐴 be the point. Then, we have the following.

Now, the above equation can be solved by rearranging it in terms of 𝑥. This gives us 90=3𝑥390+3=3𝑥93=3𝑥31=𝑥.

Thus, the solution is B, 𝑥=31.

Having seen the possible relations between points and circles in the plane, another natural question to ask is, how do lines relate to circles? Consider the diagram below.

As before, there exist three distinct possibilities in terms of how a line can interact with a circle. We can have

  1. a secant to a circle that intersects it twice (e.g., 𝐿 at points 𝐴 and 𝐵)
  2. a tangent to a circle that intersects it once (e.g., 𝐿 at point 𝐶)
  3. a line that is outside the circle completely (e.g., 𝐿).

Much like with a single point, it is possible to determine which of the above classifications a line belongs to by considering its distance from the center of the circle. However, how does one properly define the distance from a single point to a line? Let us recall the following definition.

Definition: Distance from a Point to a Line

The distance from a point 𝐴 to a line 𝐿 is the shortest distance possible from 𝐴 to any point 𝐵𝐿. This is equal to the length of the perpendicular line segment that joins 𝐴 to the nearest point on the line.

Using this definition, let us find the distance between lines 𝐿, 𝐿, and 𝐿 and the center of circle 𝑀 in the above example.

Here, we observe that, compared to the radius of the circle 𝑟, 𝑀𝐴<𝑟, 𝑀𝐵=𝑟, and 𝑀𝐶>𝑟. Just as we did before for single points, we can generalize this to a rule for any line and its classification as a secant, tangent, or line lying outside the circle.

Rule: Distance of Lines from the Center of a Circle

For a circle with center 𝑀 and radius 𝑟, and a line 𝐿, where 𝐴𝐿 is the closest point to 𝑀,

  • if 𝑀𝐴<𝑟, 𝐿 is a secant to the circle;
  • if 𝑀𝐴=𝑟, 𝐿 is a tangent to the circle;
  • if 𝑀𝐴>𝑟, 𝐿 is outside the circle.

Let us consider an example of an application of the above rule.

Example 2: Determining Whether a Line Is a Secant or a Tangent or Is Outside the Circle

Circle 𝑀 has a radius of 65. Suppose 𝐴 is on a line 𝐿 and 𝑀𝐴 is perpendicular to 𝐿. If 2𝑀𝐴56=18, what can be said of how 𝐿 lies with respect to the circle?

  1. 𝐿 is secant to circle 𝑀.
  2. 𝐿 is tangent to circle 𝑀.
  3. 𝐿 is outside of circle 𝑀.

Answer

Recall that if we compare the distance from the center of circle 𝑀 to line 𝐿 with the radius of the circle, we can determine whether 𝐿 is a secant or a tangent or if it is outside the circle.

In particular, the distance from 𝑀 to 𝐿 is defined by the length of the perpendicular line segment that joins 𝑀 to 𝐿. Since 𝑀𝐴 is perpendicular to 𝐿, it therefore measures the distance from 𝑀 to 𝐿.

Now, 𝑀𝐴 satisfies the equation 2𝑀𝐴56=18.

We can solve this to find 𝑀𝐴 by rearranging the equation: 2𝑀𝐴=18+562𝑀𝐴=74𝑀𝐴=37.

Finally, let us compare 𝑀𝐴 to the radius. Since the radius is 65 and 𝑀𝐴=37, we have 𝑀𝐴<𝑟.

We illustrate this below.

Since 𝑀𝐴<𝑟, we can conclude that, by definition, the solution is 𝐴, that is, 𝐿 is a secant to circle 𝑀.

One final thing to note on this topic is that because a tangent always intersects a circle at a single point on its circumference, and this point is the closest point on the tangent to the center, the perpendicular line segment between a tangent and the center is always a radius of the circle. As this is an important property, we will state it below.

Property: Tangents to a Circle and Radii

For a circle with center 𝑀, with a tangent 𝐿 to the circle at point 𝐴, 𝑀𝐴 is perpendicular to the tangent (i.e., 𝑀𝐴𝐿), and 𝑀𝐴 is a radius of the circle.

We note that this property goes both ways; if a line is perpendicular to a radius of the circle and intersects that radius at a point on the circumference, then it must be a tangent of the circle.

In the example below, we will use what we have learned to calculate the angles between lines and circles.

Example 3: Finding the Measure of an Angle Using the Relationship between Tangents, Radii, and Angles in a Semicircle

Given that 𝐵𝐶 is a tangent to the circle with center 𝑀 and 𝑚𝐴𝑀𝐷=97, find 𝑚𝐶𝐵𝐷.

Answer

It is always helpful to begin by putting the information we have on the diagram.

Here, angle 𝐶𝐵𝐷 is labeled 𝜃 since it is the angle we need to calculate.

We start by recognizing that 𝐴𝐵 is a straight line, and the angles on a straight line add up to 180, so 𝑚𝐵𝑀𝐷+𝑚𝐴𝑀𝐷=180𝑚𝐵𝑀𝐷+97=180𝑚𝐵𝑀𝐷=18097𝑚𝐵𝑀𝐷=83.

We also notice that 𝑀𝐵 and 𝑀𝐷 are radii of the circle (as is 𝑀𝐴, although we do not need that here). All radii of a circle have the same length; thus, 𝑀𝐵=𝑀𝐷. Let us put this information on the diagram.

Consider triangle 𝐵𝑀𝐷. Since two sides of 𝐵𝑀𝐷 are equal in length, it is an isosceles triangle. This tells us that the two remaining angles are equal to each other. Let us say 𝑚𝐷𝐵𝑀=𝑚𝐵𝐷𝑀=𝜑. We also know that the angles of a triangle add up to 180. Thus, we have 2𝜑+83=1802𝜑=180832𝜑=97𝜑=48.5.

Let us mark this on the diagram once more.

Now, finally, to work out 𝜃, we make use of the property that tangents to the circle have with respect to radii. Since 𝐵𝐶 is a tangent to the circle and 𝑀𝐵 is a radius of the circle, angle 𝑚𝐶𝐵𝑀 is therefore a right angle. Thus, we have 𝜃+48.5=90𝜃=9048.5𝜃=41.5.

Therefore, the answer is 𝑚𝐶𝐵𝐷=41.5.

So far, we have seen how points can interact with circles and how lines can interact with circles. What about circles with other circles? On a basic level, two distinct circles can intersect at two points, at one point, or not at all. However, there are additional scenarios that arise when one circle is contained within the other circle. Let us investigate these one by one.

Let us suppose we have two distinct circles: one with center 𝑀 and radius 𝑟 and one with center 𝑀 and radius 𝑟. Let 𝑀𝑀 be the distance between the two centers. Then, we have the following cases.

Case 1: Circles apart from Each Other

If 𝑀𝑀>𝑟+𝑟, then the circles are apart from each other. They do not intersect and neither one is inside the other.

Case 2: Circles Touching Externally

If 𝑀𝑀=𝑟+𝑟, then the circles intersect at one point, 𝐴, on the outside of both circles. We note that there is a tangent to both circles at 𝐴.

Case 3: Circles Intersecting at Two Points

Suppose 𝑟𝑟 (if 𝑟>𝑟, we can relabel the circles). Then, if 𝑟𝑟<𝑀𝑀<𝑟+𝑟, the two circles intersect at two points, 𝐴 and 𝐵.

Additionally, we note that this also includes the case where the center 𝑀 is inside the circle centered at 𝑀, which happens when 𝑀𝑀<𝑟. We illustrate this below.

Case 4: Circles Touching Internally

If 𝑟>𝑟 and 𝑀𝑀=𝑟𝑟, then the circles intersect at one point, 𝐴, on the inside of the circle centered at 𝑀. As in case 2, there is a tangent to both circles at point 𝐴. Note that we cannot take 𝑟=𝑟 in this case, since the circles would overlap and not be distinct.

Case 5: One Circle inside Another

In the final case, if 𝑟>𝑟 and 𝑀𝑀<𝑟𝑟, then one circle is inside the other, and the circles do not intersect.

Note that if 𝑀=𝑀 (i.e., the circles have the same center), then the two circles are concentric. This is a special case of one circle being inside another that we show below.

In all five cases, the important thing is to compare the distance 𝑀𝑀 of the centers of the circles from one another to their radii.

Let us summarize the above information as a rule.

Rule: Distance of Circles from Each Other

Suppose we have two circles, one with center 𝑀 and radius 𝑟 and one with center 𝑀 and radius 𝑟. Suppose also that 𝑟𝑟. Then,

RequirementResult
𝑀𝑀>𝑟+𝑟The circles are apart.
𝑀𝑀=𝑟+𝑟The circles touch each other externally.
𝑟𝑟<𝑀𝑀<𝑟+𝑟The circles intersect at two points.
𝑀𝑀=𝑟𝑟The circles touch each other internally.
𝑀𝑀<𝑟𝑟One circle is inside the other.

Let us consider a concrete instance of the above rule in the following example.

Example 4: Finding the Distance between Two Circles given Their Radii and How They Interact

Suppose we have two circles, one with center 𝑀 and radius 𝑟=7 and one with center 𝑀 and radius 𝑟=4. Given that the circles intersect at two distinct points, which of the following is the correct range of values for the length 𝑀𝑀?

  1. 3<𝑀𝑀<11
  2. 4<𝑀𝑀<7
  3. 3<𝑀𝑀
  4. 𝑀𝑀<11
  5. 𝑀𝑀<3

Answer

To start off, it is helpful to draw a diagram showing roughly what is happening. Using the given radii of the circles, 𝑟=7 and 𝑟=4, the fact that 𝑀𝑀 is the distance between them, and that they intersect at two different points (say 𝐴 and 𝐵), we draw the following.

We can see that the possible values that the distance 𝑀𝑀 can take are dependent upon the radii. On one hand, the circles need to not be too far apart, since then they would only intersect once or not at all. Specifically, we require 𝑀𝑀<𝑟+𝑟<7+4<11.

On the other hand, we also need the circles not to be too close, since then one would be completely inside the other. This means 𝑀𝑀>𝑟𝑟>74>3.

Thus, in order for the two circles to intersect twice, we combine the two above requirements to find that 𝑀𝑀 must lie within the following values: 3<𝑀𝑀<11.

For our final example, let us consider a case where we have two circles interacting with each other and we have to find the angles involved.

Example 5: Finding the Length of a Tangent to Two Circles given Their Radii

Given that 𝑀𝐶=16cm and 𝑁𝐵=13cm, find the length of 𝐶𝐵.

Answer

Let us start by sketching a version of the diagram where we have written in some of the information we have been given or can easily figure out. For starters, we know that 𝑀𝐶 and 𝑀𝐴 are both radii of 𝑀, and 𝑁𝐵 and 𝑁𝐴 are both radii of 𝑁, so we can mark them as 16 cm and 13 cm respectively. Aditionally, we can see that 𝐶𝐵 is a tangent to both circles, and hence it forms right angles with the radii at points 𝐵 and 𝐶. Let us draw this below.

We have also labeled the distance 𝐶𝐵 as 𝑥, since it is the distance we want to find. Now, one way to find 𝑥 is to form a rectangle by adding the point 𝐷 to 𝑀𝐶, such that 𝐵𝐶𝐷𝑁 is a rectangle. This forms a right triangle above it, 𝐷𝑀𝑁, which we can use to find the value of 𝑥 using the Pythagorean theorem. We show this below.

Here, we have formed a rectangle of length 𝑥 and height 13 cm, and a right triangle of height 1613=3cmcmcm and hypotenuse 16+13=29cmcmcm. Then, we can use the Pythagorean theorem 𝑎+𝑏=𝑐 to find 𝑥 as follows: 𝑥+3=29𝑥+9=841𝑥=832𝑥=832.cm

Finally, we can use the property of surds that 𝑎𝑏=𝑎𝑏 to simplify the answer: 𝑥=64×13𝑥=6413𝑥=813.cmcmcm

So, in conclusion, 𝐶𝐵=813cm.

Let us recap some of the key points we have learned in this explainer.

Key Points

  • For a circle with center 𝑀 and radius 𝑟, and a general point 𝑃,
    • if 𝑀𝑃<𝑟, then 𝑃 is inside the circle;
    • if 𝑀𝑃=𝑟, then 𝑃 is on the circle;
    • if 𝑀𝑃>𝑟, then 𝑃 is outside the circle.
  • For a circle with center 𝑀 and radius 𝑟, and a line 𝐿, where 𝐴𝐿 is the closest point to 𝑀 (i.e., where 𝑀𝐴𝐿),
    • if 𝑀𝐴<𝑟, then 𝐿 is a secant to the circle;
    • if 𝑀𝐴=𝑟, then 𝐿 is a tangent to the circle;
    • if 𝑀𝐴>𝑟, then 𝐿 is outside the circle.
  • For a circle with center 𝑀, with a tangent 𝐿 to the circle at point 𝐴, 𝑀𝐴 is perpendicular to the tangent (i.e., 𝑀𝐴𝐿), and 𝑀𝐴 is a radius of the circle.
  • For two circles with centers 𝑀 and 𝑀 and radiii 𝑟 and 𝑟, with 𝑟𝑟, we have the following possibilities.
    RequirementResult
    𝑀𝑀>𝑟+𝑟The circles are apart.
    𝑀𝑀=𝑟+𝑟The circles touch each other externally.
    𝑟𝑟<𝑀𝑀<𝑟+𝑟The circles intersect at two points.
    𝑀𝑀=𝑟𝑟The circles touch each other internally.
    𝑀𝑀<𝑟𝑟One circle is inside the other.

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