To find the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex] when point [tex]\( R \)[/tex] divides the line segment [tex]\( \overline{PQ} \)[/tex] in the ratio 1:3, given that the [tex]\( x \)[/tex]-coordinate of [tex]\( R \)[/tex] is -1 and the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex] is -3, we can use the section formula.
The section formula states that if a point divides a line segment joining two points in a given ratio, then the coordinates of the dividing point can be determined as follows:
[tex]\[ x = \frac{m_2 \cdot x_1 + m_1 \cdot x_2}{m_1 + m_2} \][/tex]
where:
- [tex]\( x \)[/tex] is the [tex]\( x \)[/tex]-coordinate of the dividing point,
- [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] are the [tex]\( x \)[/tex]-coordinates of the endpoints,
- [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the segments into which the original line segment is divided.
In this case:
- The ratio is 1:3, so [tex]\( m_1 = 1 \)[/tex] and [tex]\( m_2 = 3 \)[/tex].
- The [tex]\( x \)[/tex]-coordinate of [tex]\( R \)[/tex] (the dividing point) is -1.
- The [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex] is -3.
- We need to find the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex], which we'll denote as [tex]\( x_Q \)[/tex].
Let's apply the section formula to solve for [tex]\( x_Q \)[/tex]:
[tex]\[ -1 = \frac{3 \cdot (-3) + 1 \cdot x_Q}{1 + 3} \][/tex]
Simplify the equation:
[tex]\[ -1 = \frac{-9 + x_Q}{4} \][/tex]
To solve for [tex]\( x_Q \)[/tex], multiply both sides of the equation by 4:
[tex]\[ -1 \cdot 4 = -9 + x_Q \][/tex]
[tex]\[ -4 = -9 + x_Q \][/tex]
Add 9 to both sides to isolate [tex]\( x_Q \)[/tex]:
[tex]\[ -4 + 9 = x_Q \][/tex]
[tex]\[ 5 = x_Q \][/tex]
So, the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex] is [tex]\( 5 \)[/tex].
The correct answer is:
C. 5
