To determine the ratio \( \frac{AP}{BP} \), we first need to find the coordinates of point \( P \) given that its x-coordinate is 5. We can find the y-coordinate of point \( P \) by determining the equation of the line that passes through points \( A(2, 3) \) and \( B(10, 4) \).
Step 1: Find the slope of the line \( AB \)
The slope \( m \) of the line through \( A \) and \( B \) is calculated as follows:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 3}{10 - 2} = \frac{1}{8} \]
Step 2: Write the equation of the line
Using point-slope form for the line passing through point \( A(2, 3) \):
\[ y - 3 = \frac{1}{8}(x - 2) \]
Expanding this equation:
\[ y - 3 = \frac{1}{8}x - \frac{2}{8} \] \[ y = \frac{1}{8}x + 3 - \frac{1}{4} \] \[ y = \frac{1}{8}x + \frac{11}{4} \]
Step 3: Find the y-coordinate of point \( P \)
Now substitute \( x = 5 \) into the line equation:
\[ y = \frac{1}{8}(5) + \frac{11}{4} \] \[ y = \frac{5}{8} + \frac{11}{4} \] Convert \( \frac{11}{4} \) to have a common denominator:
\[ \frac{11}{4} = \frac{22}{8} \] Thus,
\[ y = \frac{5}{8} + \frac{22}{8} = \frac{27}{8} \]
So point \( P \) is \( P(5, \frac{27}{8}) \).
Step 4: Calculate lengths \( AP \) and \( BP \)
Using the distance formula, calculate \( AP \) and \( BP \).
For \( AP \):
\[ AP = \sqrt{(5 - 2)^2 + \left(\frac{27}{8} - 3\right)^2} \] Calculate \( \frac{27}{8} - 3 = \frac{27}{8} - \frac{24}{8} = \frac{3}{8} \):
\[ AP = \sqrt{(3)^2 + \left(\frac{3}{8}\right)^2} = \sqrt{9 + \frac{9}{64}} = \sqrt{\frac{576}{64} + \frac{9}{64}} = \sqrt{\frac{585}{64}} = \frac{\sqrt{585}}{8} \]
For \( BP \):
\[ BP = \sqrt{(5 - 10)^2 + \left(\frac{27}{8} - 4\right)^2} \] Calculate \( \frac{27}{8} - 4 = \frac{27}{8} - \frac{32}{8} = -\frac{5}{8} \):
\[ BP = \sqrt{(-5)^2 + \left(-\frac{5}{8}\right)^2} = \sqrt{25 + \frac{25}{64}} = \sqrt{\frac{1600}{64} + \frac{25}{64}} = \sqrt{\frac{1625}{64}} = \frac{\sqrt{1625}}{8} \]
Step 5: Calculate the ratio \( \frac{AP}{BP} \)
\[ \frac{AP}{BP} = \frac{\frac{\sqrt{585}}{8}}{\frac{\sqrt{1625}}{8}} = \frac{\sqrt{585}}{\sqrt{1625}} = \sqrt{\frac{585}{1625}} \]
Step 6: Simplifying \( \frac{585}{1625} \)
Factoring:
585 is \( 3 \times 195 \) and \( 195 = 3 \times 65 = 3 \times 5 \times 13 \) thus \( 585 = 3^2 \times 5 \times 13 \)
1625 is \( 5^2 \times 65 = 5^2 \times 5 \times 13 = 5^3 \times 13 \)
Thus:
\[ \frac{3^2 \times 5 \times 13}{5^3 \times 13} = \frac{9}{25} \text{ so } \sqrt{\frac{9}{25}} = \frac{3}{5} \]
The ratio \( \frac{AP}{BP} \) is \( \frac{3}{5} \), so the correct answer is:
A. \( \frac{3}{5} \)