Question

Look at the right triangle ABC.



A student made the following chart to prove that AB2 + BC2 = AC2.

Answer 1

Statement
Justification
1. Triangle ABC is similar to triangle BDC
1. Angle ABC = Angle BCD and Angle BCA = Angle DBC
2. BC2 = AC × DC
2. BC ÷ DC = AC ÷ BC because triangle ABC is similar to triangle BDC
3. Triangle ABC is similar to triangle ABD
3. Angle ABC = Angle ADB and Angle BAC = Angle BAD
4. AB2 = AC × AD
4. AB ÷ AD = AC ÷ AB because triangle ABC is similar to triangle ABD
5. AB2 + BC2 = AC × AD + AC × DC
= AC (AD + DC)
5. Adding Statement 1 and Statement 2
6. AB2 + BC2 = AC2
6. AD + DC = AC

What is the flaw in the student’s proof?

Justification 4 should be “AB ÷ AD = AB ÷ AC because triangle ABC is similar to triangle ABD”.

Justification 1 should be “Angle ABC = Angle BDC and Angle BCA = Angle BCD”.

Justification 2 should be “BC ÷ DC = BC ÷ AC because triangle ABC is similar to triangle BDC”.

Justification 3 should be “Angle ABC = Angle BAD and Angle BAC = Angle ABD”.

Answer 2

http://learn.flvs.net/webdav/assessment_images/educator_geometry_v14/pool_Geom_3641_1000_Subtest_02_07/image0014e8c6013.jpg

Answer 3

Is it option #3?

Answer 4

May someone please help me?

Answer 5

Angle ABC = Angle BDC*
What is BC2? Is that BC^2?

Answer 6

Yes, it is the latter... BC^2. Sorry. I guess it was a typo.

Answer 7

No worries.
And as you can see the first statement in justification is fixed. That was a flaw because the two 90 degree angles are similar and not a 90 degree and a 45 degree angle.

Answer 8

Ok. Thanks!!!

Answer 9

Are there any units? I can't make assumptions for justification 2.

Answer 10

No. That's all they provided.

Answer 11

Well we can assume that length BC is one less than AC. Or is AC two units longer?

Answer 12

The look equal, but I wouldn't assume that. They don't often draw things to scale...

Answer 13

But I'd say they are equal, sine it is the angle bisector there, and the median of AC

Answer 14

We can use cosine law to find the lengths of BC and DC.

Answer 15

just for clarification
2. BC ÷ DC = AC ÷ BC because triangle ABC is similar to triangle BDC
means
length(BD) is to length(DC) as length(AC) is to length(BC)
i.e. ratios based on corresponding legs of similar triangles.

Answer 16

Yes. But we can't assume any lengths till we find the exact number which we can manipulate to justify the statement made regarding the similar triangles.

Answer 17

ok, So, we find the lengths, then do as phi suggested?

Answer 18

Yes we should. Let me get to a pleasant environment. I hate being near controversies depicted on news channels.

Answer 19

lol soo random aha.

Answer 20

Yes... me too. Always one party making unbelievable accusations against the other.... but then again, it's important, because it'll affect the people in this nation, and in other nations(maybe) as well.

Answer 21

Now, back to math.....