If (s − a) + (s − b) + (s − c) = s, then the value of \(\rm\frac{(s−a)^2+(s−b)^2+(s−c)^2+s^2}{a^2+b^2+c^2}\) will be
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Understanding the Algebraic Problem
The problem provides us with an equation involving the variables \(s\), \(a\), \(b\), and \(c\), and asks us to find the value of a specific algebraic expression involving the same variables. The given equation is \((s - a) + (s - b) + (s - c) = s\). The expression whose value we need to determine is \(\rm\frac{(s−a)^2+(s−b)^2+(s−c)^2+s^2}{a^2+b^2+c^2}\).
Simplifying the Given Equation
Let's start by simplifying the given equation:
\(\qquad (s - a) + (s - b) + (s - c) = s\)
Combine the terms on the left side:
\(\qquad s - a + s - b + s - c = s\)
\(\qquad 3s - (a + b + c) = s\)
Now, isolate the term involving \(a, b, c\) and \(s\):
\(\qquad 3s - s = a + b + c\)
\(\qquad 2s = a + b + c\)
This relationship, \(a+b+c = 2s\), is key to solving the problem.
Evaluating the Algebraic Expression
We need to find the value of the expression: \(\rm\frac{(s−a)^2+(s−b)^2+(s−c)^2+s^2}{a^2+b^2+c^2}\).
Let's focus on the numerator first: \((s−a)^2+(s−b)^2+(s−c)^2+s^2\).
Expand the squared terms using the algebraic identity \((x-y)^2 = x^2 - 2xy + y^2\):
- \((s-a)^2 = s^2 - 2as + a^2\)
- \((s-b)^2 = s^2 - 2bs + b^2\)
- \((s-c)^2 = s^2 - 2cs + c^2\)
Now, let's sum these three expanded terms:
\((s-a)^2 + (s-b)^2 + (s-c)^2 = (s^2 - 2as + a^2) + (s^2 - 2bs + b^2) + (s^2 - 2cs + c^2)\)
Group the terms:
\(= (s^2 + s^2 + s^2) - (2as + 2bs + 2cs) + (a^2 + b^2 + c^2)\)
\(= 3s^2 - 2s(a + b + c) + (a^2 + b^2 + c^2)\)
From our simplified equation, we know that \(a+b+c = 2s\). Substitute this into the expression:
\(= 3s^2 - 2s(2s) + (a^2 + b^2 + c^2)\)
\(= 3s^2 - 4s^2 + a^2 + b^2 + c^2\)
\(= -s^2 + a^2 + b^2 + c^2\)
Now, we need to add the remaining \(s^2\) term from the original numerator expression:
Numerator \(= (s-a)^2+(s-b)^2+(s-c)^2+s^2\)
Numerator \(= (-s^2 + a^2 + b^2 + c^2) + s^2\)
Numerator \(= a^2 + b^2 + c^2\)
So, the expression we need to evaluate becomes:
\(\rm\frac{a^2+b^2+c^2}{a^2+b^2+c^2}\)
Assuming that \(a^2+b^2+c^2 \neq 0\), the value of this expression simplifies to 1.
If \(a^2+b^2+c^2 = 0\), then since \(a, b, c\) are typically real numbers in such problems, this would imply \(a=0, b=0, c=0\). Substituting these into the original equation \((s-a)+(s-b)+(s-c)=s\) gives \((s-0)+(s-0)+(s-0)=s\), which is \(3s=s\), implying \(2s=0\), so \(s=0\). In this specific case (\(a=b=c=s=0\)), the expression becomes \(0/0\), which is indeterminate. However, since numerical options are provided, we assume a scenario where the denominator is not zero, leading to a definite value.
Conclusion on the Value of the Expression
Based on the simplification of the given equation and the algebraic expression, the value is found to be 1.
| Step | Calculation/Reasoning |
|---|---|
| 1 | Simplify the given equation: \((s-a)+(s-b)+(s-c)=s \implies 3s-(a+b+c)=s \implies a+b+c=2s\) |
| 2 | Expand the squared terms in the numerator: \((s-a)^2+(s-b)^2+(s-c)^2 = (s^2-2as+a^2) + (s^2-2bs+b^2) + (s^2-2cs+c^2)\) |
| 3 | Group terms: \(= 3s^2 - 2s(a+b+c) + (a^2+b^2+c^2)\) |
| 4 | Substitute \(a+b+c=2s\): \(= 3s^2 - 2s(2s) + (a^2+b^2+c^2) = 3s^2 - 4s^2 + a^2+b^2+c^2 = -s^2 + a^2+b^2+c^2\) |
| 5 | Add the remaining \(s^2\) to the numerator: \((-s^2 + a^2+b^2+c^2) + s^2 = a^2+b^2+c^2\) |
| 6 | Form the final expression: \(\rm\frac{a^2+b^2+c^2}{a^2+b^2+c^2}\) |
| 7 | Simplify (assuming denominator \(\neq 0\)): \(= 1\) |
Revision Table: Algebraic Simplification
Reviewing the key steps involved in simplifying algebraic expressions and equations:
- Always start by simplifying any given equations to establish relationships between variables.
- Expand squared or higher power terms carefully using appropriate algebraic identities.
- Collect like terms.
- Substitute known relationships between variables to simplify expressions further.
- Pay attention to potential cases where denominators might be zero, although often the context of multiple-choice options implies a non-zero denominator and a specific value.
Additional Information: Related Algebraic Concepts
This problem involves basic algebraic manipulation. Here are some related concepts:
- Algebraic Equations: Statements where two algebraic expressions are equal, used to find unknown values or relationships between variables.
- Algebraic Expressions: Combinations of variables, constants, and algebraic operations (addition, subtraction, multiplication, division, exponentiation).
- Identities: Equations that are true for all possible values of the variables involved (e.g., \((x-y)^2 = x^2 - 2xy + y^2\)).
- Substitution: Replacing a variable or expression with another equivalent variable or expression to simplify an equation or expression.
- Semi-perimeter: In geometry, if \(a, b, c\) are side lengths of a triangle, \(s = (a+b+c)/2\) is the semi-perimeter, leading to the relationship \(a+b+c=2s\). While the problem doesn't explicitly state it's a triangle, the relationship \(a+b+c=2s\) derived from the given equation is consistent with this concept.
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