A chord length 14cm is midway a radius of a circle.calculate the radius of a circle
Ah, I see you're playing a little game of hide and seek with me. Well, if the chord length is 14cm and it's right in the middle of the radius, then we can use a little mathematical trickery.
The radius of a circle is the distance from the center to any point on the circumference. Now, if the chord is right in the middle, it will also be the diameter of the circle (that's just a fancy word for a chord that passes through the center).
Think of it as two clowns balancing on a seesaw - if their weight is evenly distributed, the center of the seesaw will be right in the middle. So, the radius is half of the diameter, and if the chord is 14cm, then the diameter (and radius) of the circle would be 14cm multiplied by 2, which is 28cm.
But hey, don't take my word for it. Go measure those clowns on a real seesaw and let me know if I'm right!
To calculate the radius of the circle, you can use the Pythagorean theorem. The chord of the circle divides the radius into two segments. One of these segments forms a right triangle with the chord as its hypotenuse.
Let's denote the length of the chord as "c" and the length of one of the segments as "x." Since the chord is midway the radius, the other segment will also have the same length as "x."
According to the Pythagorean theorem, the sum of the squares of the two segments is equal to the square of the hypotenuse (chord). So we have:
x^2 + x^2 = c^2
2x^2 = c^2
Now, we can substitute the given chord length into the equation:
2x^2 = 14^2
2x^2 = 196
Dividing both sides of the equation by 2:
x^2 = 98
Taking the square root of both sides:
x = √98
x ≈ 9.899
Since x represents one of the segments of the radius, the radius of the circle will be twice this value:
radius = 2x ≈ 2 * 9.899 ≈ 19.798 cm
Therefore, the radius of the circle is approximately 19.798 cm.
To find the radius of a circle given that a chord length is midway, we can use the relationship between the chord, radius, and their perpendicular bisector.
Here's how we can solve the problem step by step:
1. Draw a circle and label the center point as O.
2. Draw a line from O to one endpoint of the chord and label it as A.
3. Draw a line from O to the midpoint of the chord and label it as M.
4. Draw a perpendicular bisector from M to the chord, intersecting it at point B.
5. Since the chord is midway the radius, OB will be equal to the radius, and MB will be half the chord length, which is 7 cm.
M M
|\ |\
| \ | \
7cm| \ 7cm 7cm| \ 7cm
| \ | \
|____\___________ A ______|____\
O O
Now, we can apply the Pythagorean theorem to find the radius. According to the theorem:
(MB)^2 + (OB)^2 = (OA)^2
Substituting the known values:
(7cm)^2 + (OB)^2 = (14cm)^2
Simplifying the equation:
49cm^2 + (OB)^2 = 196cm^2
Subtracting 49cm^2 from both sides:
(OB)^2 = 147cm^2
Finally, taking the square root of both sides to find OB (which is equal to the radius):
OB = √(147cm^2)
Simplifying:
OB ≈ 12.12 cm
Therefore, the radius of the circle is approximately 12.12 cm.
Draw a diagram. If the radius is 2r, then
r^2 + 7^2 = (2r)^2
3r^2 = 49
now finish it off