Final answer:
To find the radius of the circle, we apply the Pythagorean theorem to the right-angled triangle formed by half of the chord, the perpendicular distance from the center to the chord, and the radius of the circle. The radius is the square root of the sum of the squares of the lengths of the other two sides, resulting in approximately 17.69 cm.
Step-by-step explanation:
The student is asking how to find the radius of a circle given the length of a chord and the perpendicular distance from the chord to the center of the circle. This problem can be solved using the properties of right triangles and the Pythagorean theorem. Since the line segment from the circle's center to the midpoint of the chord forms a perpendicular bisector, two right-angled triangles are formed. Given that the length of the chord is 24 cm (which makes each half 12 cm) and the distance from the center of the circle to the chord is 13 cm, we can use these dimensions as the sides of the right triangle. Let 'r' be the radius of the circle.
The Pythagorean theorem states that in a right triangle, the sum of squares of the two shorter sides is equal to the square of the longest side (hypotenuse), which in this case would be the radius of the circle. The formula is a² + b² = c², where 'a' and 'b' are the legs of the triangle, and 'c' is the hypotenuse. So, using the given distances:
- Half the length of the chord: 12 cm
- Perpendicular distance from the center: 13 cm
- Radius of the circle: 'r'
We substitute into the Pythagorean theorem:
12² + 13² = r²
144 + 169 = r²
313 = r²
Thus, √313 = r
The radius of the circle is √313 cm, which is approximately 17.69 cm.