AB and CD are two parallel chords of a circle such that length of AB and CD are 10 cm and 24cm respectively.if the chords are opposite sides of the centre and the distance between them is 17 cm ,find the radius of the circle.
To find the radius of the circle, we can use the property that the perpendicular bisector of a chord passes through the center of the circle.
Let's denote the center of the circle as O. We know that AB and CD are parallel chords on opposite sides of the center. So, the distance between them is the perpendicular distance from the center O to the line containing AB and CD.
Given that the distance between AB and CD is 17 cm, we can draw a perpendicular line from O to the line containing AB and CD. This perpendicular line will bisect AB and CD at their midpoints.
Let's denote the midpoint of AB as M and the midpoint of CD as N. Since the lengths of AB and CD are given as 10 cm and 24 cm respectively, the distances MO and NO can be found by dividing the lengths in half. Therefore, MO = 5 cm and NO = 12 cm.
Now, we have a right-angled triangle MON with side lengths MO = 5 cm, NO = 12 cm, and MN as the unknown radius of the circle.
To find the radius MN, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Applying the Pythagorean theorem to triangle MON, we have:
MN^2 = MO^2 + NO^2
Substituting the values, we get:
MN^2 = 5^2 + 12^2
MN^2 = 25 + 144
MN^2 = 169
Taking the square root of both sides, we find:
MN = sqrt(169)
MN = 13 cm
Therefore, the radius of the circle is 13 cm.
As usual, draw a diagram. If you draw a line through the center, perpendicular to the chords, you cut them in half. The radius is the hypotenuse of two right triangles. One has a leg of 5, and the other has a leg of 12. So,
5^2 + 12^2 = r^2
12^2 + (17-x)^2 = r^2
5^2 + x^2 = 12^2 + (17-x)^2
x = 12
so, the radius is 13
Draw a diagram,the line from the centre O meet AB at E and CD at F.
Let OF be x then OE = 17-x.
Join AO and OC
In Triangle AOE,
Radius r²= 5² + (17-x)² and
In Triangle COF,
Radius r²= 12² + x² then
12² + x²= 5² + (17-x)²
144 + x² = 25 + 289 -34x + x²
34x + x² - x² = 314 - 144
34x = 170
x = 5
So, r² = 12² + 5²
r² = 144 + 25
r² = 169
r = 13 cm /-