If the area of a circle is divided by 9 what happens to the radius? What happens to the circumference?
I am stumped on this question from ICE-EM Secondary 2B. The answer in the back of the book only states both decrease by a factor of 3.
I want to know how this happens so that I can understand it and write a formula/explanation into my exercise book. Can anybody explain this to me?
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What happens to the radius and circumference of a circle if the area is divided by 9?
All circles are similar. We know that for similar figures all corresponding linear measurements are in the same ratio called the scale factor. Thus for any two circles, if we compare radii as some ratio a:b, every corresponding length in the two circles will be in the ratio a:b; this includes the diameter, circumference, every corresponding chord, etc...
Also, if two figures are similar then their corresponding areas will be in a ratio that is the square of the scale factor.
For the given problem we have a circle and another circle reduced so that its area is 1/9 the area of the preimage circle. Thus the areas are in a ratio of 1:9.
1:9 is the square of the scale factor so `a^2:b^2=1:9 ==> a:b=1:3 `
Thus the circumference and radii (and all other corresponding lengths) of the image circle are 1/3 the length of the original circle.
** Knowing this is slightly more efficient than computing the corresponding area, circumference, etc... Indeed, this relationship can help you compute an area or length that you could not get any other way.
area of a circle = `(Pi * r^2)` = A
New area =` (A/9)` =`(Pi * r^2)/9`
= `(Pi * (r/3)^2)`
so when area is divided by 9 the radius will be divided by 3
circumference of the orginal circle = `2*pi*r`
circumference of the new circle = `2*pi*r/3`
= (circumference of the orginal circle) /3
The question says that the are of the circle A is divided by 9 so it means that the are of the circle has been reduced by a factor of 9. As we know that the area of a circle is now we have A/9 so
As A = A/9
`r^2 = r^2/9`
Hence the radius is reduced by a factor of 3.
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