Finding Circle Chord Length Using Radius and Distance

The problem involves a circle where we know the radius and the distance from the center to the midpoint of a chord. We need to find the length of the chord.

Geometric Principle

A key property in circle geometry is that the line segment drawn from the center of the circle to the midpoint of a chord is perpendicular to the chord. This creates a right-angled triangle with:

• The radius ($r$) as the hypotenuse.
• The distance from the center to the chord's midpoint ($d$) as one leg.
• Half the length of the chord ($c/2$) as the other leg.

Applying the Pythagorean Theorem

We can use the Pythagorean theorem, which states $a^2 + b^2 = h^2$ for a right-angled triangle, where $a$ and $b$ are the legs and $h$ is the hypotenuse. In this case:

• $h = r = 26$ cm
• $d = 10$ cm
• $a = c/2$

Substituting these into the theorem gives:

$ (c/2)^2 + d^2 = r^2 $

Calculation Steps

1. Substitute the given values into the equation:

   $ (c/2)^2 + 10^2 = 26^2 $

2. Calculate the squares:

   $ (c/2)^2 + 100 = 676 $

3. Isolate $(c/2)^2$:

   $ (c/2)^2 = 676 - 100 $

   $ (c/2)^2 = 576 $

4. Find the value of $c/2$ by taking the square root:

   $ c/2 = \sqrt{576} $

   $ c/2 = 24 \text{ cm} $

5. Calculate the full length of the chord ($c$) by multiplying $c/2$ by 2:

   $ c = 2 \times (c/2) $

   $ c = 2 \times 24 $

   $ c = 48 \text{ cm} $

Conclusion

The length of the chord is 48 cm.