Another problem:
Do these equate, if so how:
1 - 1 / sq.rt 2 = (sq.rt 2 - 1) / sq.rt 2
I have a rectangle sides of sq.rt 2 and 1
I cut it in half to show that the ratio of the new rectangles is teh same as the original.
Here's what I did:
New rectangle is sq.rt 2 / 2 : 1
That's as far as I got.
The book says:
sq.rt 2 / 2 : 1 = 1 / sq.rt 2 : 1 = sq.rt 2 : 1
-I don't understand how they do this last line.
Thanks.
The first one.
So you cut the sqrt(2) by 1 rectangle into to equal parts. Each half is
(1/2)sqrt(2) by 1
Or, [sqrt(2)]/2 by 1
So the ratio of the new dimensions is
[sqrt(2)]/2 : 1
You want to check if that is the same as the ratio in the original, sqrt(2) : 1.
2 is sqrt(2)*sqrt(2), so,
[sqrt(2)]/2 : 1
= [sqrt(2)]/[sqrt(2) *sqrt(2)] : 1
= 1/[sqrt(2)] : 1
Multiply both by sqrt(2),
= 1 : sqrt(2) ---------------------same as the original.
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The second one.
1 - 1 / sq.rt 2 = (sq.rt 2 - 1) / sq.rt 2
Clear the fractions, multiply both sides by sqrt(2),
sqrt(2) -1 = sqrt(2) -1
Hey, one step only.
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Another way.
The RHS is (sqrt(2) -1) / sqrt(2)
So divide,
= sqrt(2)/sqrt(2) -1/sqrt(2)
= 1 - 1/sqrt(2)
That's it.
Hello, GAdams!
I have a rectangle sides of and
I cut it in half to show that the ratio of the new rectangles is the same as the original.
Here's what I did:
New rectangle is . . . . not quite correct
This is the original rectangle:The ratio of length-to-width is: .Code:* - - - - - - - - - * | | | | 1 | | 1 | | | | * - - - - - - - - - * √2
Bisect it:Code:* - - - - * - - - - * | | | | | | 1 | | | 1 | | | | | | * - - - - * - - - - * ½√2 ½√2
Each small rectangle looks like this:The ratio of length-to-width is: .Code:* - - - - - * | | | | ½√2 | | * - - - - - * 1
And we must show that the two ratios are equal.
The first ratio is: .
The second ratio is: .
. . Rationalize: .
Therefore, the ratios are equal.
Hello, GAdams!
We can't order a proportion randomly.I don't understand why in the first ratio you have used the 1 as the denominator
and then as the numerator for the second ratio.
In both ratios, it is Length-to-Width.
. . That is, .the longer side : the shorter side.
If you don't "line them up" correctly, you're asking for trouble.Code:* - - - - - - - - - * | | * - - * | | | | 40 | | 3 | | | | | | * - - - - - - - - - * * - - * 60 2
These two rectangles are similar; the side are proportional.
But not because: . . . . . which is not true.
Get the idea?
You seem to be multiplying the (1 - 1) by sq.rt 2.?
No. I multiplied the numerator 1 of the 1/1 by sqrt(2).
It is subtraction of fractions.
Example, 1/2 -3/4
Common denominator is 4.
4 divided by 2 ...[equals 2], times 1, equals 2
4 divided by 4 ...[equals 1] , times 3, equals 3
So,
1/2 -3/4
= (2 -3)/4
= -1/4
Now compare that to your surd above.