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

What are the coordinates of point \( Q \)?

A. \((25, 22)\)

B. \((25, 13)\)

C. \(\left( -\frac{22}{5}, \frac{23}{5} \right)\)

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



Answer :

GinnyAnswer
Given:
- Point \(P\) at coordinates \((-10, 3)\).
- Point \(R\) at coordinates \((4, 7)\).
- Ratio \(P R: R Q = 2:3\).

To find the coordinates of point \(Q\), we use the section formula for a point dividing a line segment internally in a given ratio. The section formula for a point dividing the segment joining \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m:n\) is:

[tex]\[ (x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \][/tex]

Given that point \(R\) divides segment \(P Q\) in the ratio \(2:3\), we need to essentially reverse-engineer the section formula to determine the coordinates of \(Q\).

Let's denote the coordinates of \(Q\) as \((x_Q, y_Q)\).

Using the coordinates of \(R\) and the ratio, the section formula gives the coordinates of \(R\):

[tex]\[ R_x = \frac{2 \cdot x_Q + 3 \cdot (-10)}{5} \][/tex]

Given \(R_x = 4\), we can set up the equation:

[tex]\[ 4 = \frac{2 x_Q - 30}{5} \][/tex]

Multiplying both sides by 5:

[tex]\[ 20 = 2 x_Q - 30 \][/tex]

Adding 30 to both sides:

[tex]\[ 50 = 2 x_Q \][/tex]

Dividing by 2:

[tex]\[ x_Q = 25 \][/tex]

Similarly, for the y-coordinate:

[tex]\[ R_y = \frac{2 \cdot y_Q + 3 \cdot 3}{5} \][/tex]

Given \(R_y = 7\), we can set up the equation:

[tex]\[ 7 = \frac{2 y_Q + 9}{5} \][/tex]

Multiplying both sides by 5:

[tex]\[ 35 = 2 y_Q + 9 \][/tex]

Subtracting 9 from both sides:

[tex]\[ 26 = 2 y_Q \][/tex]

Dividing by 2:

[tex]\[ y_Q = 13 \][/tex]

Therefore, the coordinates of point \(Q\) are \((25, 13)\).

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

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