This is talking about "similar" figures.
Two drawings are "similar" if they have the same shape but different size.
All squares are similar to all other squares.
All circles are similar to all other circles.
All equilateral triangles are similar to all other equilateral triangles.
For any other shapes (polygons that are not regular), you have to
check it out, and you have to know what you're looking for.
Here's the most important tool. You should memorize it:
If two figures are similar, then their corresponding sides
are all in the same ratio.
That means that if I have two 4-sided shapes, and one is big and
one is small but they look like the same shape, then I need to look
at side BIG-A and side little-a and compare their lengths.
They won't be the same. I can clearly see that one is big and
the other one is small. What I need to do is divide the big one
by the little one. That gives me their ratio.
-- Let's say I divide their lengths and I get 2 . (That means that
side BIG-A is double the length of side little-a.)
OK. I have the ratio of ONE pair of sides, a big one and a little one.
-- Now ... I pick another pair of sides: side BIG-B and side little-b .
I divide those, and find their ratio.
-- Then ... I pick another pair of sides: side BIG-C and side little-c .
I divide those, and find their ratio.
-- Finally ... I pick the last pair of sides: side BIG-D and side little-d .
If I find that ALL of the ratios are the same number ... every side of the
BIG figure is 2 times (double) the length of the same side of the little
figure ... THEN the two figures are 'similar'. They're really the same shape
but just different size.
Again, the Rule:
In order to be similar, the corresponding pairs of all sides
in the figures must all be in the same ratio.
Fine. Now you can solve Charlotte's problem.
I told you all that to tell you this:
Charlotte has two rectangles. She wants them to be similar.
That tells us that (little length)/(BIG LENGTH) and (little width)/(BIG WIDTH)
must be the same ratio.
Great. We know the ratio of their widths. It's 3 / 7.5 = 0.4 .
Every dimension of the little rectangle has to be 0.4 of the
same dimension of the BIG one.
So she has to make the little length 0.4 of the BIG LENGTH .
The BIG LENGTH is 22.5 .
The little length has to be (0.4 x 22.5) = 9 cm .
If you're confused, then stop here.
If you're still with me, I'll show you another thing about 'similar' figures:
Up to now, we only looked at the ratio of a BIG SIDE to a little side,
and we said they have to be the same number.
It turns out that IF the figures are similar, then the corresponding ratios
of two parts in the same figure are also equal. What does that mean ?
I mean like the ratio of (BIG LENGTH)/(BIG WIDTH) and the ratio
of (little length)/(little width).
Look at the two rectangles Charlotte has now:
Length: 22.5
Width: 7.5 ratio of length/width
in the same rectangle = 3
Length: 9
Width: 3 ratio of length/width
in the same rectangle = 3 yay !
That also shows that the two rectangles are 'similar'.
