Question

The Pythagorean Theorem is a2+b2=c2a
2
+b
2
=c
2
. Determine if the equations shown below are true variations of the Pythagorean Theorem.
True
Not true
b
=
c
2
−
a
2
b=
c
2
−a
2

​



b
=
c
−
a
b=c−a


c
=
a
+
b
c=a+b


a
=
b
2
−
c
2
a=
b
2
−c
2

​



a
=
c
2
−
b
2
a=
c
2
−b
2

​



c
=
a
+
b
c=
a+b
​

Answer 1

Let's evaluate each of the equations one by one to see if they represent true variations of the Pythagorean Theorem, which states that \( a^2 + b^2 = c^2 \).

1. **\( b = c^2 - a^2 \)**
**Not True.**
The correct expression for \( b \) in terms of \( a \) and \( c \) is \( b^2 = c^2 - a^2 \), or \( b = \sqrt{c^2 - a^2} \).

2. **\( b = c - a \)**
**Not True.**
This equation does not relate to the Pythagorean Theorem. Instead, it represents a linear relationship not grounded in the context of right triangles.

3. **\( c = a + b \)**
**Not True.**
This also does not represent a relationship from the Pythagorean Theorem. The Pythagorean Theorem involves squares of the sides, not their sums.

4. **\( a = b^2 - c^2 \)**
**Not True.**
The proper relation involving \( a \) here would be \( a^2 = c^2 - b^2 \), or \( a = \sqrt{c^2 - b^2} \).

5. **\( a = c^2 - b^2 \)**
**Not True.**
Similarly, for \( a \), the correct equation would be \( a^2 = c^2 - b^2 \), or \( a = \sqrt{c^2 - b^2} \).

6. **\( c = a + b \)** (This seems to repeat the earlier equation #3)
**Not True.**
Same reasoning as before; this does not correspond to the Pythagorean Theorem.

In summary, none of the equations provided are true variations of the Pythagorean Theorem. All are incorrect representations of the relationships defined by \( a^2 + b^2 = c^2 \).