Certainly!
Given the equation:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
This is a well-known mathematical relationship known as the Pythagorean theorem, which applies to right-angled triangles. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse ([tex]\(c\)[/tex]) is equal to the sum of the squares of the lengths of the other two sides ([tex]\(a\)[/tex] and [tex]\(b\)[/tex]).
### Step-by-Step Solution:
1. Identify the Variables:
- [tex]\(a\)[/tex]: one leg of the right-angled triangle
- [tex]\(b\)[/tex]: the other leg of the right-angled triangle
- [tex]\(c\)[/tex]: the hypotenuse of the right-angled triangle
2. Given Equation:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
3. Explanation of the Equation:
- [tex]\(a^2\)[/tex] represents the square of one leg.
- [tex]\(b^2\)[/tex] represents the square of the other leg.
- When you sum these two squares, the result is equal to the square of the hypotenuse ([tex]\(c^2\)[/tex]).
### Hypothetical Example:
To understand this better, let's consider a numerical example:
- Suppose [tex]\(a = 3\)[/tex] and [tex]\(b = 4\)[/tex].
4. Calculate [tex]\(c\)[/tex]:
[tex]\[
c = \sqrt{a^2 + b^2}
\][/tex]
5. Substitute the values into the equation:
[tex]\[
a^2 = 3^2 = 9
\][/tex]
[tex]\[
b^2 = 4^2 = 16
\][/tex]
[tex]\[
a^2 + b^2 = 9 + 16 = 25
\][/tex]
[tex]\[
c^2 = 25 \rightarrow c = \sqrt{25} = 5
\][/tex]
So, if [tex]\(a = 3\)[/tex] and [tex]\(b = 4\)[/tex], then [tex]\(c = 5\)[/tex].
### General Conclusion:
In summary, for any right-angled triangle, the relationship given by [tex]\(a^2 + b^2 = c^2\)[/tex] holds true. This relationship can be used to find any one of the three sides if the other two are known.
