Determine if the two figures are congruent and explain your answer.

- anonymous

Determine if the two figures are congruent and explain your answer.

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- anonymous

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- anonymous

I know they ARE congruent I just need help explaining why.

- anonymous

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## More answers

- anonymous

@phi

- anonymous

@phi Please help

- phi

if you divide each figure into two triangles and show each triangle is congruent by side-side-side, to the corresponding triangle in the other figure, that would prove the figures are congruent.

- anonymous

That is what I say in my own words or do I have to do it too??

- phi

that is how to show the figures are congruent.
but then you have to do the work.
the length of each side can be found using the distance formula.
do you know how to do that ?

- anonymous

d= sqrt(x_2-x_1/y_2-y_1)?

- anonymous

I totally forgot but I know it is something like that

- phi

the idea is based on the pythagorean theorem
a^2 + b^2 = c^2
where a and b are the legs and c is the hypotenuse

- phi

the formula tells you how to find the length of each side using the coordinates
but in your graph, we can just count.

- phi

For example the leg from point A to point D
|dw:1438103882199:dw|

- anonymous

ye but how do we do it on mines?

- phi

it looks like if we slide the figure to try to overlay the other figure, side AD matches up with the side GH
so find the length of GH. if you get sqrt(17) then they are the same length

- phi

make a triangle using side GH
it has height 1 and width 4

- anonymous

|dw:1438104168143:dw|

- anonymous

Now what?

- phi

yes. now pick another side in the first figure. side CD
and compare it to side EH in the other figure.

- anonymous

In the problem or example?

- phi

The examples are from your problem. In other words, we have already shown AD and GH have the same size.
now we do CD and EH

- anonymous

CD= Length 1; Width 4
EH= Length -4; Width 1
Am I right?

- phi

yes, but we don't care if it's -4. when we find the length -4*-4 = 16 (always positive)
so if they have the same numbers 1 and 4 then the lengths will end up being the same

- anonymous

CD 1*1 + 4*4 = 16+1 = 17 = sqrt17
EH 1*1 +-4*-4 = 16 +1 = 17 = sqrt17
So segment CD and EH are congruent.

- phi

yes

- anonymous

So now what do I do?

- phi

I would do BC and FE
(I mentally shifted the figures so they lay on top of each other to decide which legs should match up)

- anonymous

BC: Height 2; Width -1
FE: Height 1; Width 2
Right?

- phi

yes

- anonymous

BC 2*2 + -1*-1 = 2+1 = 3 = sqrt3
FE 1*1 + 2*2 = 1 + 2 = 3 = sqrt3
Segment BC IS congruent to Segment FE
Did I do it right?

- phi

they are congruent, but you check 2*2 is 4 (not 2)

- anonymous

oh yeah oops
sqrt5
sqrt5

- anonymous

So now what do I do?

- phi

what two sides are left to check?

- anonymous

FG and AB

- phi

yes

- anonymous

AB: Height 1; Width -2
FG: Height 2; Width 1
Right?

- phi

yes

- anonymous

AB 1*1 + -2*-2 = 1+4 = 5 = sqrt5
FG 2*2 + 1*1 = 4+1 = 5 = sqrt5
Segment AB IS congruent to Segment FE
Right?

- phi

yes.
I think there is a theorem that says if all 4 sides are congruent , then the two figures are congruent. But we don't have to use it. if we show that BD = HF (the diagonals)
then that shows the triangles match by SSS, and I know that definitely is true.

- anonymous

So I just say that they are congruent because BD=HF by SSS or do I write what we did?

- phi

BD=HF by doing the same thing: both are 3 width and 3 height
so each is sqrt(18) long and therefore congruent.

- phi

so we know that triangle BCD can be laid directly on top of triangle FEH
because they are congruent by SSS
we also know triangle BAD can be laid on top of triangle FGH because they are congruent.
when we do that , we see that the shape ABCD lies directly on top of GFEH
proving they are congruent.

- anonymous

Oh okay thank you @phi I appreciate you making me able to understand this.

- phi

yw

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