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Sagot :
To solve the problem, let's break it down step-by-step:
### Step 1: Understanding the Formula
The formula we need to use is:
[tex]\[ R = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1 \][/tex]
where:
- [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] are the coordinates of points [tex]\( Q \)[/tex] and [tex]\( S \)[/tex] respectively.
- [tex]\( m \)[/tex] and [tex]\( n \)[/tex] represent the ratio in which point [tex]\( R \)[/tex] divides the line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex].
### Step 2: Identifying the Given Values
From the problem:
- The coordinate of [tex]\( Q \)[/tex] is [tex]\( -8 \)[/tex] (i.e., [tex]\( x_1 = -8 \)[/tex]).
- The coordinate of [tex]\( S \)[/tex] is [tex]\( 12 \)[/tex] (i.e., [tex]\( x_2 = 12 \)[/tex]).
- The ratio is [tex]\( 4:1 \)[/tex], meaning [tex]\( m = 4 \)[/tex] and [tex]\( n = 1 \)[/tex].
### Step 3: Substituting the Values into the Formula
Plugging these values into the formula:
[tex]\[ R = \left(\frac{4}{4+1}\right)(12 - (-8)) + (-8) \][/tex]
Simplifying inside the parentheses first:
[tex]\[ R = \left(\frac{4}{5}\right)(12 + 8) + (-8) \][/tex]
[tex]\[ R = \left(\frac{4}{5}\right)(20) + (-8) \][/tex]
[tex]\[ R = \left(\frac{4 \times 20}{5}\right) + (-8) \][/tex]
[tex]\[ R = \left(\frac{80}{5}\right) + (-8) \][/tex]
[tex]\[ R = 16 + (-8) \][/tex]
[tex]\[ R = 8 \][/tex]
### Step 4: Verifying Which Expression Matches
Now let's examine each given expression to determine which one matches our simplified formula:
Expression 1:
[tex]\[ \left(\frac{1}{1+4}\right)(12 - (-8)) + (-8) \quad \Rightarrow \quad \left(\frac{1}{5}\right)(12 + 8) + (-8) \quad \Rightarrow \quad \left(\frac{1}{5}\right)(20) + (-8) \quad \Rightarrow \quad 4 - 8 \quad \Rightarrow \quad -4 \][/tex]
Expression 2:
[tex]\[ \left(\frac{4}{4+1}\right)^2(12 - (-8)) + (-8) \quad \Rightarrow \quad \left(\frac{4}{5}\right)^2(12 + 8) + (-8) \quad \Rightarrow \quad \left(\frac{16}{25}\right)(20) + (-8) \quad \Rightarrow \quad \frac{320}{25} - 8 \quad \Rightarrow \quad 12.8 - 8 \quad \Rightarrow \quad 4.8 \][/tex]
Expression 3:
[tex]\[ \left(\frac{4}{4+1}\right)(-8 - 12) + 12 \quad \Rightarrow \quad \left(\frac{4}{5}\right)(-8 - 12) + 12 \quad \Rightarrow \quad \left(\frac{4}{5}\right)(-20) + 12 \quad \Rightarrow \quad -16 + 12 \quad \Rightarrow \quad -4 \][/tex]
Expression 4:
[tex]\[ \left(\frac{4}{1+4}\right)(-8 - 12) + 12 \quad \Rightarrow \quad \left(\frac{4}{5}\right)(-20) + 12 \quad \Rightarrow \quad -16 + 12 \quad \Rightarrow \quad -4 \][/tex]
The expression that correctly matches our calculation:
[tex]\[ \left(\frac{4}{4+1}\right)(12 - (-8)) + (-8) \][/tex]
Thus, the correct expression is:
[tex]\[ \left(\frac{4}{4+1}\right)(12 - (-8)) + (-8) \][/tex]
Therefore, none of the given expressions in the options exactly follow the correct solution for finding [tex]\( R \)[/tex]. However, if we assume the typo in option 1 as [tex]\(\left(\frac{4}{4+1}\right)(12 - (-8)) + (-8)\)[/tex], the resultant coordinate balances precisely with our answer [tex]\(R = 8\)[/tex].
### Step 1: Understanding the Formula
The formula we need to use is:
[tex]\[ R = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1 \][/tex]
where:
- [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] are the coordinates of points [tex]\( Q \)[/tex] and [tex]\( S \)[/tex] respectively.
- [tex]\( m \)[/tex] and [tex]\( n \)[/tex] represent the ratio in which point [tex]\( R \)[/tex] divides the line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex].
### Step 2: Identifying the Given Values
From the problem:
- The coordinate of [tex]\( Q \)[/tex] is [tex]\( -8 \)[/tex] (i.e., [tex]\( x_1 = -8 \)[/tex]).
- The coordinate of [tex]\( S \)[/tex] is [tex]\( 12 \)[/tex] (i.e., [tex]\( x_2 = 12 \)[/tex]).
- The ratio is [tex]\( 4:1 \)[/tex], meaning [tex]\( m = 4 \)[/tex] and [tex]\( n = 1 \)[/tex].
### Step 3: Substituting the Values into the Formula
Plugging these values into the formula:
[tex]\[ R = \left(\frac{4}{4+1}\right)(12 - (-8)) + (-8) \][/tex]
Simplifying inside the parentheses first:
[tex]\[ R = \left(\frac{4}{5}\right)(12 + 8) + (-8) \][/tex]
[tex]\[ R = \left(\frac{4}{5}\right)(20) + (-8) \][/tex]
[tex]\[ R = \left(\frac{4 \times 20}{5}\right) + (-8) \][/tex]
[tex]\[ R = \left(\frac{80}{5}\right) + (-8) \][/tex]
[tex]\[ R = 16 + (-8) \][/tex]
[tex]\[ R = 8 \][/tex]
### Step 4: Verifying Which Expression Matches
Now let's examine each given expression to determine which one matches our simplified formula:
Expression 1:
[tex]\[ \left(\frac{1}{1+4}\right)(12 - (-8)) + (-8) \quad \Rightarrow \quad \left(\frac{1}{5}\right)(12 + 8) + (-8) \quad \Rightarrow \quad \left(\frac{1}{5}\right)(20) + (-8) \quad \Rightarrow \quad 4 - 8 \quad \Rightarrow \quad -4 \][/tex]
Expression 2:
[tex]\[ \left(\frac{4}{4+1}\right)^2(12 - (-8)) + (-8) \quad \Rightarrow \quad \left(\frac{4}{5}\right)^2(12 + 8) + (-8) \quad \Rightarrow \quad \left(\frac{16}{25}\right)(20) + (-8) \quad \Rightarrow \quad \frac{320}{25} - 8 \quad \Rightarrow \quad 12.8 - 8 \quad \Rightarrow \quad 4.8 \][/tex]
Expression 3:
[tex]\[ \left(\frac{4}{4+1}\right)(-8 - 12) + 12 \quad \Rightarrow \quad \left(\frac{4}{5}\right)(-8 - 12) + 12 \quad \Rightarrow \quad \left(\frac{4}{5}\right)(-20) + 12 \quad \Rightarrow \quad -16 + 12 \quad \Rightarrow \quad -4 \][/tex]
Expression 4:
[tex]\[ \left(\frac{4}{1+4}\right)(-8 - 12) + 12 \quad \Rightarrow \quad \left(\frac{4}{5}\right)(-20) + 12 \quad \Rightarrow \quad -16 + 12 \quad \Rightarrow \quad -4 \][/tex]
The expression that correctly matches our calculation:
[tex]\[ \left(\frac{4}{4+1}\right)(12 - (-8)) + (-8) \][/tex]
Thus, the correct expression is:
[tex]\[ \left(\frac{4}{4+1}\right)(12 - (-8)) + (-8) \][/tex]
Therefore, none of the given expressions in the options exactly follow the correct solution for finding [tex]\( R \)[/tex]. However, if we assume the typo in option 1 as [tex]\(\left(\frac{4}{4+1}\right)(12 - (-8)) + (-8)\)[/tex], the resultant coordinate balances precisely with our answer [tex]\(R = 8\)[/tex].
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