PQ and RS are 2 parallel chords of a circle such that PQ = 8 cm & RS = 6 cm. If the chords are on the opposite sides of the centre and distance between them is 4 cm. find the radius of the circle
To find the radius of the circle, we can use the property of chords of a circle that states that if two chords of a circle are parallel, on opposite sides of the center, and have the same distance between them, then the line segment joining the midpoints of the chords passes through the center of the circle.
Let's denote the midpoint of PQ as M and the midpoint of RS as N. We also know that the distance between PQ and RS is 4 cm, so the distance between M and N is half of that, which is 2 cm. Since M and N lie on the same line segment joining the midpoints of the chords, it means they both lie on the diameter of the circle passing through the center.
Now, we have a right-angled triangle MNP, where MN is the radius of the circle and MP and NP are radii perpendicular to chords PQ and RS. We can apply the Pythagorean theorem to find the value of MN.
Let's consider triangle MNP.
Using the Pythagorean theorem, we have:
(MN)^2 = (MP)^2 + (NP)^2
Since MP and NP are radii of the circle, they have the same length, which is half the length of the chords.
MP = PQ/2 = 8/2 = 4 cm
NP = RS/2 = 6/2 = 3 cm
Substituting these values into our equation, we get:
(MN)^2 = (4)^2 + (3)^2
(MN)^2 = 16 + 9
(MN)^2 = 25
Taking the square root of both sides, we find:
MN = √25
MN = 5 cm
Therefore, the radius of the circle is 5 cm.
r^2-x^2 = 4^2
r^2-(4-x)^2 = 3^2
now just solve for r