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How To Find The Radius Of A Circle With Two Points

How To Find The Radius Of A Circle With Two Points

4 min read 27-11-2024
How To Find The Radius Of A Circle With Two Points

How to Find the Radius of a Circle Given Two Points

Determining the radius of a circle knowing only two points on its circumference might seem like an impossible task. However, with a little geometry and algebra, it's entirely solvable, though not with a single, straightforward formula. The process involves several steps, and the difficulty increases if you only have the coordinates of the two points. Let's break down the different scenarios and methods.

Scenario 1: You know the distance between the two points and the location of the center.

This is the simplest case. Imagine you have two points, A and B, on a circle's circumference. You also know the coordinates of the circle's center, O. The radius is simply the distance from the center to either point.

  • Method: Use the distance formula. The distance formula, derived from the Pythagorean theorem, calculates the distance between two points (x₁, y₁) and (x₂, y₂) in a Cartesian coordinate system:

    d = √[(x₂ - x₁)² + (y₂ - y₁)²]

    Where 'd' represents the distance. Apply this formula to calculate the distance between the center O (x₀, y₀) and either point A or B (x₁, y₁). This distance 'd' is the radius of the circle.

  • Example: Let's say the center of the circle is at O(3, 4), and point A is at (6, 8).

    Radius = √[(6 - 3)² + (8 - 4)²] = √(3² + 4²) = √(9 + 16) = √25 = 5

    The radius of the circle is 5 units.

Scenario 2: You know the distance between the two points and the circle's diameter.

If you know the distance between the two points and the diameter of the circle, finding the radius is trivial.

  • Method: The radius is half the diameter.

    Radius = Diameter / 2

  • Example: Let's say the distance between points A and B is 8 units, and the diameter of the circle is 10 units.

    Radius = 10 / 2 = 5

    The radius of the circle is 5 units. Note that this scenario implies that the line segment AB is a chord of the circle, and it's not necessarily a diameter.

Scenario 3: You only know the coordinates of the two points.

This is the most challenging scenario. We need to find the center of the circle first before calculating the radius. This involves several steps:

  • Step 1: Find the midpoint of the line segment connecting the two points.

    The midpoint represents the center of the circle if the line segment connecting the two points is a diameter. However, if it's just a chord, the center will lie somewhere on the perpendicular bisector of the chord.

    Let's say the two points are A(x₁, y₁) and B(x₂, y₂). The midpoint M(xₘ, yₘ) is calculated as:

    xₘ = (x₁ + x₂) / 2 yₘ = (y₁ + y₂) / 2

  • Step 2: Find the slope of the line segment AB.

    The slope (m) of the line passing through points A and B is:

    m = (y₂ - y₁) / (x₂ - x₁)

  • Step 3: Find the slope of the perpendicular bisector.

    The perpendicular bisector of AB is a line that passes through the midpoint M and is perpendicular to AB. The slope (mₚ) of the perpendicular bisector is the negative reciprocal of the slope of AB:

    mₚ = -1 / m

  • Step 4: Find the equation of the perpendicular bisector.

    Using the point-slope form of a line equation, we can write the equation of the perpendicular bisector:

    y - yₘ = mₚ(x - xₘ)

  • Step 5: This is where it gets tricky. We need more information. Knowing only two points on a circle's circumference is insufficient to uniquely define the circle. Infinitely many circles can pass through those two points. To proceed, we need additional information, such as another point on the circle or the circle's equation.

  • Step 6 (If additional information is available): Use the additional information to find the center of the circle. This could involve solving a system of equations or using geometric properties.

Scenario 4: Using the equation of a circle.

If you know the equation of the circle (in the standard form: (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius), and you have two points on the circle, you can:

  • Method 1: Substitute the coordinates of one of the points into the equation. This will give you an equation with only r unknown. Solve for r.

  • Method 2: Substitute both points into the equation. This creates a system of two equations with two unknowns (h and k). Solve for h and k, then substitute into the equation to find r. This method is typically more complex.

Illustrative Example (Scenario 3 – Incomplete):

Let's say we have points A(1, 2) and B(5, 6).

  1. Midpoint: M = ((1+5)/2, (2+6)/2) = (3, 4)

  2. Slope of AB: m = (6-2)/(5-1) = 4/4 = 1

  3. Slope of perpendicular bisector: mₚ = -1/1 = -1

  4. Equation of perpendicular bisector: y - 4 = -1(x - 3) => y = -x + 7

We cannot proceed further without more information. We know the center lies on the line y = -x + 7, but it could be anywhere on that line.

Conclusion:

Finding the radius of a circle given only two points on its circumference is not directly solvable with a single formula. The solution requires additional information, either the location of the center, the diameter, or another point on the circle. Understanding the geometry and utilizing the distance formula and the equation of a circle are crucial for navigating these different scenarios. The problem highlights the importance of having sufficient constraints to solve geometric problems uniquely.

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