Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To determine the [tex]\( x \)[/tex]-coordinate of point [tex]\( Q \)[/tex], let's use the section formula. Given that point [tex]\( R \)[/tex] divides the segment [tex]\( \overline{PQ} \)[/tex] in the ratio [tex]\( 1:3 \)[/tex], and we know the [tex]\( x \)[/tex]-coordinate of [tex]\( R \)[/tex] is -1, and the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex] is -3, we are to find the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex].
The section formula for a point [tex]\( R \)[/tex] that divides the segment [tex]\( \overline{PQ} \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ R_x = \frac{m \cdot Q_x + n \cdot P_x}{m + n} \][/tex]
In this problem:
- [tex]\( R_x = -1 \)[/tex]
- [tex]\( P_x = -3 \)[/tex]
- [tex]\( \frac{RQ}{PQ} = \frac{1}{3} \)[/tex] which gives [tex]\( m = 1 \)[/tex] and [tex]\( n = 3 \)[/tex]
Substituting the known values into the formula, we get:
[tex]\[ -1 = \frac{1 \cdot Q_x + 3 \cdot (-3)}{1 + 3} \][/tex]
First, simplify the equation on the right-hand side:
[tex]\[ -1 = \frac{Q_x - 9}{4} \][/tex]
Now, eliminate the fraction by multiplying both sides by 4:
[tex]\[ -1 \cdot 4 = Q_x - 9 \][/tex]
This gives:
[tex]\[ -4 = Q_x - 9 \][/tex]
To solve for [tex]\( Q_x \)[/tex], add 9 to both sides of the equation:
[tex]\[ -4 + 9 = Q_x \][/tex]
[tex]\[ Q_x = 5 \][/tex]
Thus, the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex] is [tex]\( 5 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{5} \][/tex]
The section formula for a point [tex]\( R \)[/tex] that divides the segment [tex]\( \overline{PQ} \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ R_x = \frac{m \cdot Q_x + n \cdot P_x}{m + n} \][/tex]
In this problem:
- [tex]\( R_x = -1 \)[/tex]
- [tex]\( P_x = -3 \)[/tex]
- [tex]\( \frac{RQ}{PQ} = \frac{1}{3} \)[/tex] which gives [tex]\( m = 1 \)[/tex] and [tex]\( n = 3 \)[/tex]
Substituting the known values into the formula, we get:
[tex]\[ -1 = \frac{1 \cdot Q_x + 3 \cdot (-3)}{1 + 3} \][/tex]
First, simplify the equation on the right-hand side:
[tex]\[ -1 = \frac{Q_x - 9}{4} \][/tex]
Now, eliminate the fraction by multiplying both sides by 4:
[tex]\[ -1 \cdot 4 = Q_x - 9 \][/tex]
This gives:
[tex]\[ -4 = Q_x - 9 \][/tex]
To solve for [tex]\( Q_x \)[/tex], add 9 to both sides of the equation:
[tex]\[ -4 + 9 = Q_x \][/tex]
[tex]\[ Q_x = 5 \][/tex]
Thus, the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex] is [tex]\( 5 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{5} \][/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.