If 2s = a + b + c, then what is s2 + (s - a)(s - b) + (s - b)(s - c) + (s - c)(s - a) equal to ?
ab + bc + ca
Understanding the Problem: Algebraic Simplification
The question asks us to simplify a given algebraic expression involving variables \(s, a, b,\) and \(c\), with the condition that \(2s = a + b + c\). We need to find the value of the expression \(s^2 + (s - a)(s - b) + (s - b)(s - c) + (s - c)(s - a)\) in terms of \(a, b,\) and \(c\).
Step-by-Step Algebraic Solution
We are given the relationship:
\[2s = a + b + c\]From this, we can express \(s\) as:
\[s = \frac{a + b + c}{2}\]Now, let's expand each term in the expression we need to simplify:
\((s - a)(s - b)\)
Using the distributive property (FOIL method):
\[(s - a)(s - b) = s^2 - sb - sa + ab = s^2 - s(a + b) + ab\]\((s - b)(s - c)\)
Similarly:
\[(s - b)(s - c) = s^2 - sc - sb + bc = s^2 - s(b + c) + bc\]\((s - c)(s - a)\)
And:
\[(s - c)(s - a) = s^2 - sa - sc + ca = s^2 - s(c + a) + ca\]Now, substitute these expanded forms back into the original expression:
\(s^2 + (s - a)(s - b) + (s - b)(s - c) + (s - c)(s - a)\)
Substitute the expanded terms:
\[s^2 + (s^2 - s(a + b) + ab) + (s^2 - s(b + c) + bc) + (s^2 - s(c + a) + ca)\]Combine like terms:
\[s^2 + s^2 - s(a + b) + ab + s^2 - s(b + c) + bc + s^2 - s(c + a) + ca\] \[(s^2 + s^2 + s^2 + s^2) + (-s(a + b) - s(b + c) - s(c + a)) + (ab + bc + ca)\] \[4s^2 - s((a + b) + (b + c) + (c + a)) + ab + bc + ca\] \[4s^2 - s(a + b + b + c + c + a) + ab + bc + ca\] \[4s^2 - s(2a + 2b + 2c) + ab + bc + ca\] \[4s^2 - 2s(a + b + c) + ab + bc + ca\]Now, use the given relationship \(2s = a + b + c\). We can substitute \(2s\) for \((a + b + c)\) in the expression:
\[4s^2 - 2s(2s) + ab + bc + ca\] \[4s^2 - 4s^2 + ab + bc + ca\] \[0 + ab + bc + ca\] \[ab + bc + ca\]The expression simplifies to \(ab + bc + ca\).
Result of Algebraic Simplification
The value of \(s^2 + (s - a)(s - b) + (s - b)(s - c) + (s - c)(s - a)\) is \(ab + bc + ca\), given that \(2s = a + b + c\).
| Term | Expansion |
|---|---|
| \((s-a)(s-b)\) | \(s^2 - s(a+b) + ab\) |
| \((s-b)(s-c)\) | \(s^2 - s(b+c) + bc\) |
| \((s-c)(s-a)\) | \(s^2 - s(c+a) + ca\) |
Revision Table: Key Concepts
Reviewing the key steps and concepts used in this algebraic simplification problem:
- Understanding the given condition \(2s = a + b + c\) and its implication for \(s\).
- Expanding product terms like \((s - a)(s - b)\) using algebraic identities or distributive property.
- Combining like terms after substituting expanded forms.
- Using the given condition to substitute and simplify the resulting expression.
- Basic algebraic manipulation skills are crucial for solving such problems.
Additional Information: Related Concepts
The relationship \(s = \frac{a+b+c}{2}\) is commonly used in Heron's formula for the area of a triangle, where \(s\) is the semi-perimeter and \(a, b, c\) are the side lengths. This problem provides a useful algebraic identity related to this semi-perimeter concept.
The terms \((s-a), (s-b), (s-c)\) can also be expressed as:
- \(s - a = \frac{a+b+c}{2} - a = \frac{a+b+c - 2a}{2} = \frac{b+c-a}{2}\)
- \(s - b = \frac{a+b+c}{2} - b = \frac{a+b+c - 2b}{2} = \frac{a+c-b}{2}\)
- \(s - c = \frac{a+b+c}{2} - c = \frac{a+b+c - 2c}{2} = \frac{a+b-c}{2}\)
While we used the expansion method directly with \(s\) in the main solution, using these forms can sometimes be helpful for alternative approaches or related problems.
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