Select the correct answer.

Point [tex]R[/tex] divides [tex]\overline{EF}[/tex] in the ratio [tex]1:5[/tex]. If the coordinates of [tex]E[/tex] and [tex]F[/tex] are [tex](4,8)[/tex] and [tex](11,4)[/tex], respectively, what are the coordinates of [tex]R[/tex] to two decimal places?

A. [tex](4.66, 7.62)[/tex]

B. [tex](6, 6.86)[/tex]

C. [tex](5.17, 7.33)[/tex]

D. [tex](9.83, 4.67)[/tex]



Answer :

To determine the coordinates of point [tex]\( R \)[/tex] which divides the line segment [tex]\(\overline{EF}\)[/tex] in the ratio [tex]\(1:5\)[/tex], given [tex]\( E = (4, 8) \)[/tex] and [tex]\( F = (11, 4) \)[/tex], we can use the section formula.

The section formula for a point [tex]\( R = (x, y) \)[/tex] dividing a line segment [tex]\(\overline{EF}\)[/tex] in the ratio [tex]\( \frac{m}{n} \)[/tex] is given by:

[tex]\[ R = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right) \][/tex]

Here, [tex]\( (x_1, y_1) \)[/tex] are the coordinates of [tex]\( E \)[/tex], [tex]\( (x_2, y_2) \)[/tex] are the coordinates of [tex]\( F \)[/tex], [tex]\( m = 1 \)[/tex], and [tex]\( n = 5 \)[/tex].

Substituting the given points and the ratio into the formula:

[tex]\[ x_R = \frac{1 \cdot 11 + 5 \cdot 4}{1 + 5} \][/tex]
[tex]\[ x_R = \frac{11 + 20}{6} \][/tex]
[tex]\[ x_R = \frac{31}{6} \][/tex]
[tex]\[ x_R \approx 5.17 \][/tex]

Similarly,

[tex]\[ y_R = \frac{1 \cdot 4 + 5 \cdot 8}{1 + 5} \][/tex]
[tex]\[ y_R = \frac{4 + 40}{6} \][/tex]
[tex]\[ y_R = \frac{44}{6} \][/tex]
[tex]\[ y_R \approx 7.33 \][/tex]

Therefore, the coordinates of point [tex]\( R \)[/tex] to two decimal places are [tex]\((5.17, 7.33)\)[/tex].

The correct answer is:
C. [tex]\((5.17, 7.33)\)[/tex]