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

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

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

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Other answers:

@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.
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 ?
d= sqrt(x_2-x_1/y_2-y_1)?
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|
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
|dw:1438104168143:dw|
Now what?
  • phi
yes. now pick another side in the first figure. side CD and compare it to side EH in the other figure.
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
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
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
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)
BC: Height 2; Width -1 FE: Height 1; Width 2 Right?
  • phi
yes
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)
oh yeah oops sqrt5 sqrt5
So now what do I do?
  • phi
what two sides are left to check?
FG and AB
  • phi
yes
AB: Height 1; Width -2 FG: Height 2; Width 1 Right?
  • phi
yes
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.
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.
Oh okay thank you @phi I appreciate you making me able to understand this.
  • phi
yw

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