Final answer:
To find the radius of the circle with parallel chords AB and CD, construct right triangles by drawing perpendiculars from the circle's center to the chords and apply Pythagorean theorem to solve for the distance d and then the radius r.
Step-by-step explanation:
Finding the Radius of a Circle with Parallel Chords
To find the radius of a circle when given parallel chords on opposite sides of the center and the distance between them, we can create right triangles to use the Pythagorean theorem. The chords AB and CD are parallel and equidistant from the center O, creating two right triangles. Chord AB is 10 cm and CD is 24 cm, with a separation of 17 cm. We will denote the radius of the circle as r, the distance from the center to chord AB as d, and to CD as d+17 cm since they are on opposite sides.
Draw perpendiculars from O to AB and CD. This will create two right triangles, OAB and OCD. For triangle OAB, we have:
OA = r (radius)
OB = 10/2 = 5 cm (half of AB)
Using Pythagoras' theorem:
r2 = d2 + 52
For triangle OCD:
OC = r (radius)
OD = 24/2 = 12 cm (half of CD)
Using Pythagoras' theorem:
r2 = (d + 17)2 + 122
Now we have two equations:
r2 = d2 + 25
r2 = (d + 17)2 + 144
Since both expressions are equal to r2, we can set them equal to each other:
d2 + 25 = (d + 17)2 + 144
Solve for d to find the distance from the center to chord AB. Substitute back into any one of the right triangle equations and solve for r to find the radius of the circle.