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| 11 Theory slides |
| 7 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Try your knowledge on these topics.
In the diagram below, can the larger circle be mapped onto the smaller circle? What transformations are involved?
Consider two concentric circles with common center O and radii 4 and 6.
Let A and B be two points on the smaller and larger circles, respectively, collinear with O. Also, consider the ray through A and B, with endpoint O.
Since 4×23=6, point B is the image of point A after a dilation with center O and scale factor 23. Likewise, if any point on the smaller circle is dilated through O by a factor of 23, its image will fall on the larger circle. Hence, the circles are related by a dilation, and so, are similar. The same argument can be used for any two concentric circles. This leads to the following conclusion.
Concentric circles are similar.
Dilate the concentric circles below so that they map onto each other.
Given two concentric circles on a coordinate plane, Zosia is trying to determine the transformation that maps the smaller circle onto the larger circle.
Consider the fact that the circles have different radii.
Note that the circles have different radii. The radius of the smaller circle is 3, and the radius of the larger circle is 6.
Now, given a different pair of concentric circles, Zosia wants to find the transformation that maps the larger circle onto the smaller circle.
Consider the fact that the circles have different radii.
It can be seen in the diagram that the radius of the smaller circle is 3, and the radius of the larger circle is 9.
Non-concentric circles are similar.
With this information and knowing that any two concentric circles are also similar, a more general statement can be made.
All circles are similar.
Two non-concentric circles can be mapped onto each other through either a translation or by a combination of a translation and a dilation. Therefore, non-concentric circles are similar. Translate and dilate the non-concentric circles below so that they map onto each other.
Diego has been asked to identify the transformation that maps the circle on the left onto the circle on the right.
Consider the fact that the circles have the same radius.
It can be seen in the diagram that the radius of both circles is 3.
Finally, Diego wants to identify the combination of transformations that maps the smaller circle onto the larger circle.
The radius of the smaller circle is 3, and the radius of the larger circle is 4.5. By which number does 3 need to be multiplied to obtain 4.5?
It can be seen in the diagram that the smaller circle has a radius of 3 and its center at the point (-3,-2). Furthermore the larger circle is centered at (5.5,2.5) and its radius is 4.5
In this lesson, it has been proven that all circles are similar by using similarity transformations. Therefore, any theorem that is valid for one circle, is also valid for all circles. This can be exemplified by the following theorem.
If two inscribed angles of a circle intercept the same arc, then they are congruent. |
Which of the following circles are similar to iv?
Similarity means that two figures have the same shape. Basically what this tells us is that its only their size that differs. For that to be the case, the ratio of each figure's corresponding sides must be the same. Below we see an example.
However, in circles there is only one dimension that decides the size — the radius. Therefore, we will always be able to map any two circles onto each other by dilating the radius of one of them. It is formalized by the Similar Circles Theorem that all circles are similar.
Is there enough information to claim that ⊙A∼⊙B. Explain your reasoning.
The symbol ~ means similarity. Therefore, the statement is telling us that circle A is similar to circle B. A ~ B ⇓ A is similar to B By the Similar Circles Theorem, we know that all circles are, in fact, similar. We can prove this by transforming one of the circles until it maps onto the other. In this case, the circles have the same radius. Therefore, all we need to do is translate one circle until the centers overlap.
Consider ⊙A and ⊙B.
The scale factor needs to dilate ⊙ A such that it maps onto ⊙ B is the ratio of the radius of A to the radius of B. Therefore, let's first measure the radii of the two circles.
As we can see, ⊙ A and ⊙ B has a radius of 5 and 2 units respectively. Therefore, to dilate A so that it maps onto ⊙ B, we have to multiply its radius by a scale factor of 25.
This time, we want to dilate ⊙ B until it maps onto ⊙ A. From Part A, we know the radii of both circles. Since we want B to map onto A, we must multiply the radius of B by a scale factor of 52.