Question

example 3
refer to the coordinate grid.

1. find point z on rs such that the ratio of rz to zs is 1:3.
2. find point g on ab such that the ratio of ag to gb is 3:2.
3. find point e on uv such that the ratio of ue to ev is 3:4.

refer to the coordinate grid.

1. find point x on ab such that the ratio of ax to xb is 1:3.
2. find point y on cd such that the ratio of dy to yc is 2:1.
3. find point z on ef such that the ratio of ez to zf is 2:3.

example 4

1. city planning the united states capitol is located at (2, - 4) on a coordinate grid. the white house is located at (-10, 16) on the same coordinate grid. find two points on the straight line between the united states capitol and the white house such that the ratio is 1:3.
2. maps leila is walking from the park at point p to a restaurant at point r. she wants to stop for a break when the distance she has traveled and the distance she has left to travel has a ratio of 3:5. at which point should leila stop for her break?

Explanation:

Step1: Recall the section - formula

If a point \(P(x,y)\) divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) 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 9

Let \(R(x_1,y_1)\) and \(S(x_2,y_2)\) be the endpoints of the line - segment \(RS\). Since the ratio \(RZ:ZS = 1:3\), \(m = 1\) and \(n = 3\). Then the \(x\) - coordinate of \(Z\) is \(x_Z=\frac{1\times x_S+3\times x_R}{1 + 3}\) and the \(y\) - coordinate of \(Z\) is \(y_Z=\frac{1\times y_S+3\times y_R}{1 + 3}\).

Step3: Solve problem 10

For the line - segment \(AB\) with \(AG:GB=3:2\), \(m = 3\) and \(n = 2\). The \(x\) - coordinate of \(G\) is \(x_G=\frac{3\times x_B+2\times x_A}{3 + 2}\) and the \(y\) - coordinate of \(G\) is \(y_G=\frac{3\times y_B+2\times y_A}{3 + 2}\).

Step4: Solve problem 11

For the line - segment \(UV\) with \(UE:EV = 3:4\), \(m = 3\) and \(n = 4\). The \(x\) - coordinate of \(E\) is \(x_E=\frac{3\times x_V+4\times x_U}{3 + 4}\) and the \(y\) - coordinate of \(E\) is \(y_E=\frac{3\times y_V+4\times y_U}{3 + 4}\).

Step5: Solve problem 12

For the line - segment \(AB\) with \(AX:XB = 1:3\), \(m = 1\) and \(n = 3\). The \(x\) - coordinate of \(X\) is \(x_X=\frac{1\times x_B+3\times x_A}{1 + 3}\) and the \(y\) - coordinate of \(X\) is \(y_X=\frac{1\times y_B+3\times y_A}{1 + 3}\).

Step6: Solve problem 13

For the line - segment \(CD\) with \(DY:YC = 2:1\), \(m = 2\) and \(n = 1\). The \(x\) - coordinate of \(Y\) is \(x_Y=\frac{2\times x_C+1\times x_D}{2 + 1}\) and the \(y\) - coordinate of \(Y\) is \(y_Y=\frac{2\times y_C+1\times y_D}{2 + 1}\).

Step7: Solve problem 14

For the line - segment \(EF\) with \(EZ:ZF = 2:3\), \(m = 2\) and \(n = 3\). The \(x\) - coordinate of \(Z\) is \(x_Z=\frac{2\times x_F+3\times x_E}{2 + 3}\) and the \(y\) - coordinate of \(Z\) is \(y_Z=\frac{2\times y_F+3\times y_E}{2 + 3}\).

Step8: Solve problem 15

Let \(A(2,-4)\) be the location of the United States Capitol and \(B(-10,16)\) be the location of the White House. For the ratio \(1:3\), \(m = 1\) and \(n = 3\).
The \(x\) - coordinate of the first point \(P_1\) is \(x_1=\frac{1\times(-10)+3\times2}{1 + 3}=\frac{-10 + 6}{4}=-1\).
The \(y\) - coordinate of \(P_1\) is \(y_1=\frac{1\times16+3\times(-4)}{1 + 3}=\frac{16-12}{4}=1\).
For the other case (swapping \(m\) and \(n\) to get the other point on the line - segment), \(m = 3\) and \(n = 1\).
The \(x\) - coordinate of the second point \(P_2\) is \(x_2=\frac{3\times(-10)+1\times2}{3 + 1}=\frac{-30 + 2}{4}=-7\).
The \(y\) - coordinate of \(P_2\) is \(y_2=\frac{3\times16+1\times(-4)}{3 + 1}=\frac{48-4}{4}=11\).

Step9: Solve problem 16

Let \(P(x_1,y_1)\) be the park and \(R(x_2,y_2)\) be the restaurant. Since the ratio of the distance traveled to the distance left is \(3:5\), the ratio of the point that divides the line - segment \(PR\) is \(m = 3\) and \(n = 5\).
Use the section formula \(x=\frac{3\times x_R+5\times x_P}{3 + 5}\) and \(y=\frac{3\times y_R+5\times y_P}{3 + 5}\).

Since the coordinates of the endpoints are not given for problems 9 - 14 in the question (only the ratio and the line - segments are mentioned), we can't give numerical answers. For problem 15, the two points are \((-1,1)\) and \((-7,11)\). For problem 16, we need the coordinates of \(P\) and \(R\) to get a numerical answer.

Answer:

For problem 15, the two points on the line between the Capitol and the White House with ratio \(1:3\) are \((-1,1)\) and \((-7,11)\). For problems 9 - 14 and 16, numerical answers cannot be provided without the coordinates of the endpoints of the line - segments.