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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]


Sagot :

To determine the coordinates of point [tex]\( R \)[/tex] that divides the line segment [tex]\( \overline{EF} \)[/tex] in the ratio [tex]\( 1:5 \)[/tex], we use the section formula. The section formula for dividing a line segment in the ratio [tex]\( m:n \)[/tex] is given by:

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

Here, point [tex]\( E \)[/tex] has coordinates [tex]\( (x_1, y_1) = (4, 8) \)[/tex] and point [tex]\( F \)[/tex] has coordinates [tex]\( (x_2, y_2) = (11, 4) \)[/tex]. The ratio [tex]\( 1:5 \)[/tex] means [tex]\( m=1 \)[/tex] and [tex]\( n=5 \)[/tex].

Substituting these values into the section formula:

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

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

Thus, the coordinates of point [tex]\( R \)[/tex] are approximately [tex]\((5.17, 7.33)\)[/tex].

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