- Read the three positions from left to right on the graph. - Compare the length from \(A\) to \(P\) with the length from \(P\) to \(B\). - Keep the ratio in the order requested, then simplify it.

1. From the graph, \(A=(0, 0)\), \(P=(2, 0)\), and \(B=(8, 0)\). 2. Therefore, \(AP=2\) units and \(PB=6\) units. 3. The ratio is \(2:6=1:3\).

\(1:3\)

- Count the equal grid intervals between each pair of labeled points. - For \(AP:PB\), compare the length from \(A\) to \(P\) with the length from \(P\) to \(B\). - For \(AP:AB\), compare \(AP\) with the entire segment.

1. Count the grid intervals from \(A\) to \(P\): \(AP=2\) units. 2. Count the grid intervals from \(P\) to \(B\): \(PB=3\) units. Therefore, \(AP:PB=2:3\). 3. The whole segment has \(AB=5\) units, so \(AP:AB=2:5\).

\(AP:PB=2:3\) \(AP:AB=2:5\)

- Read the distances between the x-coordinates. - Which ratio compares the two pieces, and which compares a piece with the entire segment?

1. The two parts have lengths \(AP=3\) and \(PB=9\), so \(AP:PB=3:9=1:3\). 2. The whole segment is \(AB=12\), so \(PA:AB=3:12=1:4\). 3. Part-to-part and part-to-whole ratios are different.

\(AP:PB=1:3\), while \(PA:AB=1:4\). The first compares two parts; the second compares one part with the whole segment.

- How many equal-sized ratio parts make the entire segment? - What is the difference between a ratio of two parts and a fraction of the whole? - Once you know the fraction of the whole, how does it affect the horizontal change from \(A\) to \(B\)?

1. The ratio has \(1+2=3\) equal parts altogether, and \(AP\) is one of them. Thus \(AP=\frac{1}{3}AB\). 2. From \(A\) to \(B\) is \(9\) units to the right. One-third of that change is \(3\) units. 3. Starting at \((-1, 4)\) and moving \(3\) units right gives \(P=(2, 4)\).

\(\frac{1}{3}\) of the segment; \(P=(2, 4)\).

- Add the two ratio parts to determine what fraction of the whole segment lies from \(A\) to \(P\). - Find the total horizontal and vertical changes from \(A\) to \(B\). - Apply the same fraction to both coordinate changes before starting from \(A\).

1. The ratio \(2:3\) has \(5\) total parts, so \(P\) is \(\frac{2}{5}\) of the way from \(A\) to \(B\). 2. The coordinate change from \(A\) to \(B\) is \((10, 10)\). Taking \(\frac{2}{5}\) of that change gives \((4, 4)\). 3. Starting at \(A\), \(P=(-3+4, 2+4)=(1, 6)\).

\(P=(1, 6)\)

- Convert the part-to-part ratio into the fraction of the entire segment measured from \(A\). - Find the displacement vector from \(A\) to \(B\). - Take the required fraction of that displacement and add it to the starting endpoint.

1. The ratio \(2:3\) means that \(P\) is \(\frac{2}{5}\) of the way from \(A\) to \(B\). 2. The displacement from \(A\) to \(B\) is \((7, 5)\), so \(\frac{2}{5}(7, 5)=(\frac{14}{5}, 2)\). 3. Add this displacement to \(A\): \((-1, 3)+(\frac{14}{5}, 2)=(\frac{9}{5}, 5)\).

\(P=(\frac{9}{5}, 5)\)

- Translate the ratio into a fraction of the whole segment measured from \(A\). - Compare each candidate point with the corresponding fraction of the displacement from \(A\) to \(B\). - Be careful about the order of the two parts in the ratio.

1. From the graph, \(A=(0, 0)\), \(B=(10, 5)\), \(P=(4, 2)\), and \(Q=(6, 3)\). 2. A \(2:3\) division places the point \(\frac{2}{5}\) of the way from \(A\) to \(B\). 3. \(\frac{2}{5}(10, 5)=(4, 2)\), which is the position of \(P\). 4. Therefore, \(P\) gives \(AP:PB=2:3\). Point \(Q\) is \(\frac{3}{5}\) of the way from \(A\), so it gives the reversed ratio \(3:2\).

\(P\). It is \(\frac{2}{5}\) of the way from \(A\) to \(B\), so \(AP:PB=2:3\).

- Turn the part-to-part ratio into a fraction of the entire path from \(A\). - Find the signed change in each coordinate from \(A\) to \(B\). - Add the correct fraction of those changes to the starting point.

1. The whole directed change from \(A\) to \(B\) is \((-4-6, -10-5)=(-10, -15)\). 2. The ratio \(2:3\) places \(P\) two-fifths of the way from \(A\) to \(B\). 3. Two-fifths of the change is \((-4, -6)\), giving \(P=(6-4, 5-6)=(2, -1)\).

\(P=(2, -1)\)

- Find the two coordinate-change vectors along the segment. - Compare matching horizontal or vertical changes, keeping the same direction. - Reduce the resulting part-to-part ratio.

1. The change from \(A\) to \(P\) is \((6, 4)\), and from \(P\) to \(B\) is \((9, 6)\). 2. Both changes have the same direction, and corresponding coordinate changes are in the ratio \(6:9=4:6=2:3\). 3. Therefore, \(AP:PB=2:3\).

\(2:3\)

- How could you compare the two pieces made by each marked point? - Use the coordinate changes from \(A\) to a candidate and from that candidate to \(B\). - Would the desired ratio change if you measured both pieces using the same scale factor?

1. From \(A=(-4, -2)\) to \(B=(8, 4)\), the total change is \((12, 6)\). One-third of that is \((4, 2)\). 2. Adding \((4, 2)\) to \(A\) gives \((0, 0)\), which is \(P\). The remaining change to \(B\) is \((8, 4)\), twice as large in each coordinate. 3. Point \(Q=(2, 1)\) is halfway from \(A\) to \(B\): both coordinate changes are \((6, 3)\). Thus \(AQ:QB=1:1\), not \(1:2\).

\(P=(0, 0)\) works: \(AP:PB=1:2\). Point \(Q=(2, 1)\) instead gives \(AQ:QB=1:1\).

- Read the coordinates of all three collinear points from the graph. - Compare the coordinate changes from \(A\) to \(P\) and from \(P\) to \(B\). - The common direction factor cancels when the two segment lengths are put in a ratio.

1. From the graph, \(A=(1, -2)\), \(P=(7, 4)\), and \(B=(11, 8)\). 2. From \(A\) to \(P\), the coordinate change is \((6, 6)\), so \(AP=6\sqrt{2}\). 3. From \(P\) to \(B\), the coordinate change is \((4, 4)\), so \(PB=4\sqrt{2}\). 4. Therefore, \(AP:PB=6\sqrt{2}:4\sqrt{2}=3:2\).

\(AP:PB=3:2\)

- Convert each part-to-part ratio into a fraction of the whole segment measured from \(A\). - Apply each fraction to the same coordinate change from \(A\) to \(B\). - After finding \(P\) and \(Q\), use the midpoint formula twice and compare the results.

a) From the graph, \(A=(-6, 0)\) and \(B=(6, 6)\). Point \(P\) is \(\frac{1}{3}\) of the way from \(A\) to \(B\), so \(P=(-2, 2)\). Point \(Q\) is \(\frac{2}{3}\) of the way from \(A\) to \(B\), so \(Q=(2, 4)\). b) The midpoint of \(\overline{PQ}\) is \(\left(\frac{-2+2}{2},\frac{2+4}{2}\right)=(0, 3)\). The midpoint of \(\overline{AB}\) is \(\left(\frac{-6+6}{2},\frac{0+6}{2}\right)=(0, 3)\). The two midpoints coincide.

a) \(P=(-2, 2)\), \(Q=(2, 4)\) b) Both midpoints are \((0, 3)\).

- Translate \(AP:PB=2:3\) into the fraction of the whole segment traveled from \(A\). - Compare that fraction with the endpoint weights in Priya's expression. - Check the proposed point by locating what fraction of the displacement from \(A\) to \(B\) it represents.

1. The ratio \(AP:PB=2:3\) places \(P\) \(\frac{2}{5}\) of the way from \(A\) to \(B\), so the endpoint weights are reversed: \(P=\frac{3A+2B}{5}\). 2. Equivalently, \(B-A=(15, 5)\), and \(A+\frac{2}{5}(B-A)=(-4, 2)+(6, 2)=(2, 4)\). 3. Priya's formula gives \(\frac{2A+3B}{5}=(5, 5)\), which is \(\frac{3}{5}\) of the way from \(A\) to \(B\) and therefore corresponds to the reversed ratio \(3:2\).

The weighting is reversed. The correct point is \(P=(2, 4)\), using \(P=\frac{3A+2B}{5}\).

- How far through the full segment is a point that divides it in the stated ratio? - Use both coordinate changes to place the rest stop. - For the remaining distance, what do the horizontal and vertical changes tell you?

1. The path changes by \((15, -9)\) coordinate units from \(A\) to \(B\). 2. The ratio \(2:1\) puts \(P\) two-thirds of the way from \(A\) to \(B\). Two-thirds of the change is \((10, -6)\). 3. Hence \(P=(-6+10, 4-6)=(4, -2)\). 4. From \(P\) to \(B\), the change is \((5, -3)\). The remaining length is \(10\sqrt{5^2+(-3)^2}=10\sqrt{34}\,\text{m}\approx58\,\text{m}\).

a) \(P=(4, -2)\) b) \(\approx58\,\text{m}\)

- Translate the part-to-part ratio into the fraction of the whole segment from \(A\) to \(P\). - Work backward from \(\overrightarrow{AP}\) to the full vector \(\overrightarrow{AB}\). - Add the full coordinate change to endpoint \(A\) and check that \(P\) has the required ratio.

1. The ratio \(3:2\) means \(P\) is \(\frac{3}{5}\) of the way from \(A\) to \(B\). Thus, \(\overrightarrow{AP}=\frac{3}{5}\overrightarrow{AB}\). 2. Therefore, \(\overrightarrow{AB}=\frac{5}{3}\overrightarrow{AP}\). 3. Since \(\overrightarrow{AP}=(6, -6)\), \(\overrightarrow{AB}=\frac{5}{3}(6, -6)=(10, -10)\). 4. Add this vector to \(A\): \(B=(-3+10, 4-10)=(7, -6)\).

\(B=(7, -6)\)

- Treat the two partition conditions one at a time. - First locate \(P\) as a fraction of the way from \(A\) to \(B\). - Then use \(P\) as the new starting endpoint when applying the second ratio.

1. Since \(AP:PB=2:5\), point \(P\) is \(\frac{2}{7}\) of the way from \(A\) to \(B\). This gives \(P=(-2, 2)\). 2. The displacement from \(P\) to \(B\) is \((10, 5)\). 3. Since \(PQ:QB=3:2\), point \(Q\) is \(\frac{3}{5}\) of the way from \(P\) to \(B\). 4. Thus, \(Q=(-2, 2)+\frac{3}{5}(10, 5)=(4, 5)\).

\(Q=(4, 5)\)