Question

1. using the illustrated figure, obtain the point a on the line segment that would partition the line segment into the ratio xa:ay = 5:1
2. in the given figure, find the point x on the line segment that would partition the line segment into the ratio cx:xd = 2:5
3. in the figure, the endpoints of the line segment are -9 and 13. determine the point x on the line segment that would partition the line segment into the ratio 7:11.

Explanation:

1.

Step1: Recall the section - formula for a one - dimensional line segment

If a point \(A\) divides the line segment joining \(x_1\) and \(x_2\) in the ratio \(m:n\), then the coordinate of \(A\) is given by \(A=\frac{mx_2+nx_1}{m + n}\). Here \(x_1 = 5\), \(x_2=17\), \(m = 5\), \(n = 1\).

Step2: Substitute the values into the formula

\[A=\frac{5\times17+1\times5}{5 + 1}=\frac{85 + 5}{6}=\frac{90}{6}=15\]

Step1: Use the section - formula for a one - dimensional line segment

If a point \(X\) divides the line segment joining \(x_1=-7\) and \(x_2 = 21\) in the ratio \(m:n=2:5\), then the coordinate of \(X\) is \(X=\frac{mx_2+nx_1}{m + n}\).

Step2: Substitute the values into the formula

\[X=\frac{2\times21+5\times(-7)}{2 + 5}=\frac{42-35}{7}=\frac{7}{7}=1\]

Step1: Apply the section - formula for a one - dimensional line segment

If a point \(X\) divides the line segment joining \(x_1=-9\) and \(x_2 = 13\) in the ratio \(m:n = 7:11\), then the coordinate of \(X\) is \(X=\frac{mx_2+nx_1}{m + n}\).

Step2: Substitute the values into the formula

\[X=\frac{7\times13+11\times(-9)}{7 + 11}=\frac{91-99}{18}=\frac{-8}{18}=-\frac{4}{9}\]

Answer:

15

2.