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3 years ago

I really need help with this !

Mathematics

1 answer:

user100 [1]3 years ago

8 0

It is obvious that Triangle BCD is congruent to Triangle EFG. EFG is twice the size of BCD.

Put the lengths into the pythagorean theorem equation. (the longest side is the hypotenuse)

So for Triangle BCD

3^2+b^2=5^2

(3×3)+b^2=(5×5)

9+b^2=25

So what plus 9 equals 25?

the square root of 16 is 4 so 4^2=16

9+16=25

Now since we know that Triangle EFG is twice the size of BCD remake the equation but the numbers will be:

6^2+8^2=10^2

Now check

(6×6)+(8×8)=(10×10)

36+64=100

100=100

So for Triangle EFG X=8

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Odd numbers take the form $2n-1$ , where $n\ge1$ is an integer. When $n=400$ , the last odd number would be 799. So we're adding

$S=1+3+5+\cdots+795+797+799$

By reversing the order of terms, we have

$S=799+797+795+\cdots+5+3+1$

and we can pair up terms in both sums at the same position to write

$2S=(1+799)+(3+797)+(5+795)+\cdots(795+5)+(797+3)+(799+1)$

so that we are basically adding 400 copies of 800, and from there we can find the value of the sum right away:

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###

We could also make use of the formulas,

$\displaystyle\sum_{i=1}^n1=n$

$\displaystyle\sum_{i=1}^ni=\dfrac{n(n+1)}2$

We have

$S=\displaystyle\sum_{i=1}^{400}(2i-1)=2\sum_{i=1}^{400}i-\sum_{i=1}^{400}1=400(400+1)-400=400^2=160,000$

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