The length of the common chord of two circles with radii 15 cm and 13 cm, whose centers are 14 cm apart, is 24 cm.
When two circles intersect, their common chord is the line segment connecting the two points of intersection. Determining its length involves applying fundamental geometric principles, particularly the Pythagorean theorem.
Understanding the Geometry of Intersecting Circles
Consider two circles with centers $C_1$ and $C_2$, and radii $r_1$ and $r_2$ respectively. Let the distance between their centers be $D$. When these circles intersect, they form a common chord, say $AB$.
Key geometric properties for this scenario:
- The line segment connecting the centers of the two circles ($C_1C_2$) is perpendicular to the common chord ($AB$).
- The line segment $C_1C_2$ bisects the common chord $AB$. This means if $M$ is the point where $C_1C_2$ intersects $AB$, then $AM = MB$.
These properties allow us to form two right-angled triangles, which are crucial for solving the problem.
Problem Parameters
Let's consolidate the given information:
| Parameter | Value |
|---|---|
| Radius of Circle 1 ($r_1$) | 15 cm |
| Radius of Circle 2 ($r_2$) | 13 cm |
| Distance Between Centers ($D$) | 14 cm |
Our goal is to find the length of the common chord, $AB$.
Step-by-Step Calculation of the Common Chord Length
To find the length of the common chord, we'll use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides ($a^2 + b^2 = c^2$). You can learn more about this fundamental theorem here.
Let $M$ be the midpoint of the common chord $AB$. Therefore, $AM = MB$. Let $h$ be the length of $AM$ (so the full chord length is $2h$). Let $x$ be the distance from $C_1$ to $M$, i.e., $C_1M = x$. Since the total distance between centers is $D = 14$ cm, the distance from $C_2$ to $M$ will be $C_2M = 14 - x$.
-
Formulate equations using the Pythagorean theorem:
- For Circle 1: Consider the right-angled triangle $\triangle C_1MA$.
- Hypotenuse = $r_1 = 15$ cm
- One leg = $C_1M = x$
- Other leg = $AM = h$
- Equation: $15^2 = x^2 + h^2 \implies 225 = x^2 + h^2$ (Equation 1)
- For Circle 2: Consider the right-angled triangle $\triangle C_2MA$.
- Hypotenuse = $r_2 = 13$ cm
- One leg = $C_2M = 14 - x$
- Other leg = $AM = h$
- Equation: $13^2 = (14 - x)^2 + h^2 \implies 169 = (14 - x)^2 + h^2$ (Equation 2)
- For Circle 1: Consider the right-angled triangle $\triangle C_1MA$.
-
Solve for $x$ (the distance from one center to the midpoint of the chord):
- From Equation 1, we can express $h^2 = 225 - x^2$.
- Substitute this into Equation 2:
$169 = (14 - x)^2 + (225 - x^2)$ - Expand $(14 - x)^2$:
$169 = (196 - 28x + x^2) + (225 - x^2)$ - Simplify the equation:
$169 = 196 - 28x + x^2 + 225 - x^2$
$169 = 421 - 28x$ - Isolate $28x$:
$28x = 421 - 169$
$28x = 252$ - Solve for $x$:
$x = \frac{252}{28} = 9$ cm - So, $C_1M = 9$ cm and $C_2M = 14 - 9 = 5$ cm.
-
Calculate $h$ (half the length of the common chord):
- Substitute the value of $x = 9$ cm back into Equation 1:
$225 = 9^2 + h^2$
$225 = 81 + h^2$ - Solve for $h^2$:
$h^2 = 225 - 81$
$h^2 = 144$ - Calculate $h$:
$h = \sqrt{144} = 12$ cm
- Substitute the value of $x = 9$ cm back into Equation 1:
-
Determine the full length of the common chord:
- The length of the common chord $AB$ is $2h$.
- Length of chord = $2 \times 12$ cm = 24 cm.
Practical Insights
This type of geometric problem is not just an academic exercise. It finds applications in various fields:
- Engineering Design: When designing structures or components that involve overlapping circular parts (e.g., gears, pipes, apertures), understanding the dimensions of shared regions is crucial.
- Architecture: Calculations involving intersecting circular features in building designs or urban planning might use similar principles.
- Manufacturing: Precision cutting or drilling operations often rely on exact geometric calculations to ensure parts fit together correctly.
By carefully applying the Pythagorean theorem and understanding the relationship between the centers, radii, and common chord of intersecting circles, we accurately determine the required length.
