determine the radius of a circle if a 24 inch chord is 9 inches from the center
Use the intersection chord property:
if two chords intersect, each subdivided into segments a,b and c,d respectively, then ab=cd.
Here, draw a diameter (which is also a chord of total length 2r) that intersects the given chord at right angles.
The 2 segments of the given chord are then a=b=12".
The diameter's segments are then
c=r+9 and d=r-9.
Thus (r+9)(r-9)=12²
r²-9²=12²
r²=144+81=225
r=15
15
Well, if I were to clown around with this question, I could say that the radius is so big that it can't even fit into a regular measuring tape! But let's get serious for a moment.
To determine the radius of a circle when you have a chord and its distance from the center, you can use a little bit of trigonometry. The distance from the center to the midpoint of the chord is called the perpendicular bisector, and it cuts the chord into two equal halves.
So, we have a right triangle where the hypotenuse is the radius (unknown), one leg is 9 inches (distance from the center to the chord), and the other leg is half of the chord, which is 24/2 = 12 inches.
Using the Pythagorean theorem (a^2 + b^2 = c^2) where a and b are the legs of the triangle and c is the hypotenuse (radius), we can solve for c:
9^2 + 12^2 = c^2
81 + 144 = c^2
225 = c^2
c = √225
c = 15 inches
So, the radius of the circle is 15 inches.
To determine the radius of the circle, we can use the following formula:
r = √(c^2 - d^2) / 2
Where:
r = radius of the circle
c = length of the chord
d = distance from the center to the chord
Given:
c = 24 inches
d = 9 inches
Substituting the values into the formula, we have:
r = √(24^2 - 9^2) / 2
= √(576 - 81) / 2
= √(495) / 2
≈ √(500) / 2 (Since 495 is very close to 500)
≈ √(25 * 20) / 2
≈ 5√20 / 2
≈ 5 * 2√5 / 2
≈ 5 * √5
Therefore, the radius of the circle is approximately 5√5 inches.
To determine the radius of a circle, you can use the properties of a right triangle formed by the chord and the radius.
In this case, we know the following information:
- The chord length is 24 inches.
- The distance from the chord to the center of the circle is 9 inches.
We can set up a right triangle with the chord as the base, the distance from the center to the chord as the vertical side (perpendicular), and the radius of the circle as the hypotenuse.
Let's label the radius as 'r'. The right triangle formed can be visualized as follows:
|\
| \
r| \ <- Hypotenuse (Radius)
| \
| \
|____\
c
Based on the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can write the equation:
r^2 = c^2 + (1/2 * b)^2
In this case, 'c' is the distance from the center to the chord (9 inches) and 'b' is half the length of the chord (12 inches since the chord length is 24 inches).
Let's substitute the values into the equation:
r^2 = 9^2 + (1/2 * 24)^2
r^2 = 81 + (12)^2
r^2 = 81 + 144
r^2 = 225
To solve for 'r', take the square root of both sides:
r = √225
r = 15 inches
Therefore, the radius of the circle is 15 inches.