Question

1. given $overline{pr}$ with $p(-6, -6)$ and $r(-2, 10)$, if $q$ lies on $overline{pr}$ such that the ratio of $pq$ to $pr$ is $5:8$, find the coordinates of $q$.
2. given $overline{gh}$ with $g(-8, 8)$ and $h(7, 2)$, if point $p$ divides $overline{gh}$ one - third of the way from $g$ to $h$, find the coordinates of $p$.
3. given $overline{ac}$ with $a(4, -7)$ and $c(-4, 11)$, if point $b$ divides $overline{ac}$ five - sixths of the way from $a$ to $c$, find the coordinates of $b$.
4. in a certain town, the mall is located 1 mile west and 8 miles north of the post office. the library is located 5 miles east and 2 miles north of the post office. if alanas house lies two - thirds of the way between the mall and the library, find the location of her home relative to the post office.
5. two cruise ships left the same port. after two hours, ship a is 30 miles west and 18 miles north of the port and ship b is 10 miles west and 27 miles south of the port. if there is a tug boat located one - fifth of the way from ship a to ship b, find the location of the tug boat relative to the port.

Explanation:

Step1: Recall the section - formula

If a point \(M(x,y)\) divides the line - segment joining \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\).

Step2: Solve problem 12

Given \(P(-6,-6)\) and \(R(-2,10)\) and the ratio \(PQ:PR = 5:8\), so \(m = 5\) and \(n=8 - 5=3\).
\(x_Q=\frac{5\times(-2)+3\times(-6)}{5 + 3}=\frac{-10-18}{8}=\frac{-28}{8}=-\frac{7}{2}\)
\(y_Q=\frac{5\times10+3\times(-6)}{5 + 3}=\frac{50 - 18}{8}=\frac{32}{8}=4\)

Step3: Solve problem 13

Given \(G(-8,8)\) and \(H(7,2)\), and the point \(P\) divides \(GH\) one - third of the way from \(G\) to \(H\), so \(m = 1\) and \(n = 2\).
\(x_P=\frac{1\times7+2\times(-8)}{1 + 2}=\frac{7-16}{3}=\frac{-9}{3}=-3\)
\(y_P=\frac{1\times2+2\times8}{1 + 2}=\frac{2 + 16}{3}=6\)

Step4: Solve problem 14

Given \(A(4,-7)\) and \(C(-4,11)\), and the point \(B\) divides \(AC\) five - sixths of the way from \(A\) to \(C\), so \(m = 5\) and \(n = 1\).
\(x_B=\frac{5\times(-4)+1\times4}{5 + 1}=\frac{-20 + 4}{6}=\frac{-16}{6}=-\frac{8}{3}\)
\(y_B=\frac{5\times11+1\times(-7)}{5 + 1}=\frac{55-7}{6}=\frac{48}{6}=8\)

Step5: Solve problem 15

The coordinates of the mall relative to the post - office are \((-1,8)\) and the coordinates of the library relative to the post - office are \((5,2)\).
Alana's house divides the line - segment from the mall to the library in the ratio \(2:1\).
\(x=\frac{2\times5+1\times(-1)}{2 + 1}=\frac{10 - 1}{3}=3\)
\(y=\frac{2\times2+1\times8}{2 + 1}=\frac{4 + 8}{3}=4\)

Step6: Solve problem 16

The coordinates of Ship A relative to the port are \((-30,18)\) and the coordinates of Ship B relative to the port are \((-10,-27)\).
The tug - boat divides the line - segment from Ship A to Ship B in the ratio \(1:4\).
\(x=\frac{1\times(-10)+4\times(-30)}{1 + 4}=\frac{-10-120}{5}=\frac{-130}{5}=-26\)
\(y=\frac{1\times(-27)+4\times18}{1 + 4}=\frac{-27 + 72}{5}=\frac{45}{5}=9\)

Answer:

1. \(Q(-\frac{7}{2},4)\)
2. \(P(-3,6)\)
3. \(B(-\frac{8}{3},8)\)
4. \((3,4)\) relative to the post - office
5. \((-26,9)\) relative to the port