Question

Look at the right triangle ABC.

A student made the following chart to prove that AB2 + BC2 = Statement Justificat
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

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”.