Congruence and Similarity of Circles
Two figures are similar if they both have the same shape, even if one has been enlarged or shrunk. All pairs of corresponding sides in similar figures are in the same ratio, called the scale factor.
Since this scale factor affects the length of sides, it is also called the length scale factor or linear scale factor.
You may have already noticed, but as it turns out, all circles are similar.
Circle Lengths
As we already discovered in our chapter on areas and volumes of similar figures, it's not just corresponding side lengths that are always in proportion in similar figures, but in fact any length at all.
Hence, we have the following fact about circles.
All circles are similar, so if
- circle $X$X has radius $r$r, diameter $d$d, and circumference $C$C, then
- circle $Y$Y will have radius $kr$kr, diameter $kd$kd, and circumference $kC$kC
for some length scale factor $k$k.
Circle Areas
We saw in our chapter on areas and volumes of similar figures that if two figures are similar with some length scale factor $k$k, then their areas will be related by an area scale factor $k^2$k2.
This rule carries over to circles too.
If circle $X$X has area $A$A, circle $Y$Y will have area $k^2A$k2A, where $k$k is the length scale factor.
$k^2$k2 is called the area scale factor.
Congruence of Circles
The following rule determines whether two circles are congruent.
If two circles are congruent, they will have equal:
1) Radii
2) Diameters
3) Circumferences
4) Areas
To prove that two circles are congruent, we only need to know one of these facts.
So why do we only need to know one of these facts?
Well, if you have a given height of a triangle, there are many triangles you could make that fit that height. But if you have a given radius, there is only one circle that you could make to fit that radius.
In fact, no two different circles have the same radius, diameter, circumference, or area, so we only need one fact to specify a particular circle.
Examples
Question 1
The circumference of circle A is $5\pi$5π and the area of circle B is $400\pi$400π.
By what factor does the area of circle A need to be enlarged to become congruent to circle B?
Question 2
The area of a circle is $35$35 cm2. Its radius is enlarged by a factor of $4$4 to create a larger circle.
Determine the area of the larger circle.
Question 3
In the diagram, the diameter of the circle with centre $D$D is $3$3 times as long as the diameter of the circle with centre $B$B.
If the smaller circle with centre $B$B is cut out of the larger circle, what fraction of the larger circle has been cut out?