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Consider directed line segment [tex]\( PQ \)[/tex]. Point [tex]\( P \)[/tex] is located at [tex]\((-10, 3)\)[/tex]. Point [tex]\( R \)[/tex], which is on segment [tex]\( PQ \)[/tex] and divides segment [tex]\( PQ \)[/tex] into a ratio of [tex]\( PR : RQ = 2 : 3 \)[/tex], is located at [tex]\((4, 7)\)[/tex].

What are the coordinates of point [tex]\( Q \)[/tex]?

A. [tex]\(\left(-\frac{22}{5}, \frac{23}{5}\right)\)[/tex]

B. [tex]\((25, 22)\)[/tex]

C. [tex]\((25, 13)\)[/tex]

D. [tex]\((-5, 13)\)[/tex]


Sagot :

To find the coordinates of point [tex]\(Q\)[/tex], given that point [tex]\(R\)[/tex] divides the segment [tex]\(PQ\)[/tex] in the ratio [tex]\(PR: RQ = 2: 3\)[/tex], and given the coordinates of points [tex]\(P\)[/tex] and [tex]\(R\)[/tex], we will use the section formula.

The coordinates of point [tex]\(P\)[/tex] are [tex]\((-10, 3)\)[/tex], and the coordinates of point [tex]\(R\)[/tex] are [tex]\((4, 7)\)[/tex]. Let the coordinates of point [tex]\(Q\)[/tex] be [tex]\((Q_x, Q_y)\)[/tex].

The section formula for a point dividing a line segment in a given ratio is:
[tex]\[R_x = \frac{m_1 Q_x + m_2 P_x}{m_1 + m_2}\][/tex]
[tex]\[R_y = \frac{m_1 Q_y + m_2 P_y}{m_1 + m_2}\][/tex]

We know:
[tex]\[R_x = 4, R_y = 7\][/tex]
[tex]\[P_x = -10, P_y = 3\][/tex]
[tex]\[m_1 = 2, m_2 = 3\][/tex]

First, we solve for [tex]\(Q_x\)[/tex]:
[tex]\[ 4 = \frac{2Q_x + 3(-10)}{2 + 3} \][/tex]
[tex]\[ 4 = \frac{2Q_x - 30}{5} \][/tex]
Multiplying both sides by 5:
[tex]\[ 20 = 2Q_x - 30 \][/tex]
Adding 30 to both sides:
[tex]\[ 50 = 2Q_x \][/tex]
Dividing by 2:
[tex]\[ Q_x = 25 \][/tex]

Next, we solve for [tex]\(Q_y\)[/tex]:
[tex]\[ 7 = \frac{2Q_y + 3(3)}{2 + 3} \][/tex]
[tex]\[ 7 = \frac{2Q_y + 9}{5} \][/tex]
Multiplying both sides by 5:
[tex]\[ 35 = 2Q_y + 9 \][/tex]
Subtracting 9 from both sides:
[tex]\[ 26 = 2Q_y \][/tex]
Dividing by 2:
[tex]\[ Q_y = 13 \][/tex]

Thus, the coordinates of point [tex]\(Q\)[/tex] are [tex]\((25, 13)\)[/tex].

The correct answer is:
C. [tex]\((25, 13)\)[/tex]
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