Question

select the correct answer.

point r divides \\( \overline { e f } \\) in the ratio \\( 1 : 5 \\). if the coordinates of e and f are \\( ( 4,8 ) \\) and \\( ( 11,4 ) \\), respectively, what are the coordinates of r to two decimal places?

\\( \bigcirc \\) a. \\( ( 4.66,7.62 ) \\)

\\( \bigcirc \\) b. \\( ( 6,6.86 ) \\)

\\( \bigcirc \\) c. \\( ( 5.17,7.33 ) \\)

\\( \bigcirc \\) d. \\( ( 9.83,4.67 ) \\)

Explanation:

Step1: Use the section formula

If a point \(R(x,y)\) divides the line segment joining \(E(x_1,y_1)\) and \(F(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2 + nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here \(m = 1\), \(n = 5\), \(x_1=4\), \(y_1 = 8\), \(x_2=11\), \(y_2 = 4\).

Step2: Calculate the \(x\) - coordinate

Substitute the values into the \(x\) - coordinate formula:
\(x=\frac{1\times11+5\times4}{1 + 5}=\frac{11 + 20}{6}=\frac{31}{6}\approx5.17\)

Step3: Calculate the \(y\) - coordinate

Substitute the values into the \(y\) - coordinate formula:
\(y=\frac{1\times4+5\times8}{1 + 5}=\frac{4+40}{6}=\frac{44}{6}\approx7.33\)

Answer:

C. \((5.17,7.33)\)