Calculating Circle Radius from Perpendicular Chords

This problem requires finding the radius of a circle based on the properties of two chords that intersect at right angles.

Chord Information Analysis:

• Center of the circle: O.
• Chords: AB and CD, intersecting at 90 degrees.
• Distance from O to AB ($d_1$): 3 cm.
• Distance from O to CD ($d_2$): 4 cm.

Geometric Principle for Perpendicular Chords:

When two chords intersect perpendicularly inside a circle, the radius ($r$) can be determined using their distances ($d_1$, $d_2$) from the center. The relationship is:

$r^2 = d_1^2 + d_2^2$

This formula arises from constructing a rectangle using the perpendicular distances and segments related to the chords.

Radius Calculation Steps:

1. Identify the given distances from the center to the chords: $d_1 = 3$ cm and $d_2 = 4$ cm.
2. Apply the formula relating the radius to chord distances: $r^2 = d_1^2 + d_2^2$.
3. Substitute the given values:

   $r^2 = (3 \text{ cm})^2 + (4 \text{ cm})^2$

4. Calculate the squares:

   $r^2 = 9 \text{ cm}^2 + 16 \text{ cm}^2$

5. Sum the results:

   $r^2 = 25 \text{ cm}^2$

6. Find the radius by taking the square root:

   $r = \sqrt{25 \text{ cm}^2}$

   $r = 5 \text{ cm}$

Result:

The radius of the circle is 5 cm.