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Understand the coordinate plane

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5190955
A coordinate map uses longitude as the x-coordinate and latitude as the y-coordinate. The equator is the line with latitude \(0\), and the prime meridian is the line with longitude \(0\). Ship \(A\) is where those two lines meet. Ship \(B\) is on the equator at longitude \(4\). a) Give the coordinates of \(A\). b) Give the coordinates of \(B\). c) From \(A\) to \(B\), which coordinate changes and which coordinate stays \(0\)?

Hints

- Use the statement that longitude is the first coordinate and latitude is the second. - Translate “equator” and “prime meridian” using the zero values stated in the problem. - For part c), compare the two ordered pairs coordinate by coordinate.

Solution

1. At the intersection of longitude \(0\) and latitude \(0\), Ship \(A\) is at \((0, 0)\). 2. Ship \(B\) has longitude \(4\), which is the x-coordinate, and latitude \(0\), which is the y-coordinate. Thus, \(B=(4, 0)\). 3. From \(A\) to \(B\), the x-coordinate changes from \(0\) to \(4\), while the y-coordinate stays \(0\).

Answer

a) \((0, 0)\) b) \((4, 0)\) c) The x-coordinate changes; the y-coordinate stays \(0\).
5504645
Point \(P\) is shown on the coordinate plane. The dashed guide segments show its horizontal and vertical positions. a) What are the coordinates of \(P\)? b) What does the first coordinate tell you about \(P\)? c) What does the second coordinate tell you about \(P\)?
Figure for problem 550464

Hints

- Use the guide segment that drops to the horizontal axis. - Use the guide segment that reaches toward the vertical axis. - Connect each position to the matching part of the ordered pair.

Solution

1. Point \(P\) is \(3\) units to the right of the y-axis and \(2\) units above the x-axis, so \(P=(3, 2)\). 2. The first coordinate, \(3\), gives the horizontal distance from the y-axis. 3. The second coordinate, \(2\), gives the vertical distance from the x-axis.

Answer

a) \((3, 2)\) b) It shows that \(P\) is \(3\) units to the right of the y-axis. c) It shows that \(P\) is \(2\) units above the x-axis.
5504655
Jordan says, “The point \((2, 7)\) is found by moving \(7\) units right and \(2\) units up from the origin.” Is Jordan correct? Explain how \((2, 7)\) should be located.

Hints

- Pay attention to the order of the two numbers. - Decide which number controls horizontal position. - Check which number controls vertical position.

Solution

1. In an ordered pair, the first coordinate gives the horizontal position and the second gives the vertical position. 2. Jordan reversed the two coordinates. 3. The point \((2, 7)\) is located by moving \(2\) units right and \(7\) units up from the origin.

Answer

No. Move \(2\) units right and \(7\) units up.
5505235
Look at the coordinate plane. a) What is the name of the point where the x-axis and y-axis meet? b) What are the coordinates of that point? c) To locate \((4, 2)\), how far do you move from the origin in the horizontal direction, and then in the vertical direction?
Figure for problem 550523

Hints

- Find the place where the two axes cross. - Think about what coordinates describe that crossing point. - Read an ordered pair from left to right.

Solution

1. The x-axis and y-axis meet at the origin. 2. The origin has coordinates \((0, 0)\). 3. For \((4, 2)\), move \(4\) units to the right and then \(2\) units up.

Answer

a) The origin b) \((0, 0)\) c) \(4\) units right, then \(2\) units up
5188055
Points \(C(2, 8)\) and \(D(6, 8)\) lie on line \(h\). a) How far is line \(h\) from the x-axis? b) How far is point \(C\) from the y-axis? c) Find the length of \(\overline{CD}\).

Hints

- Compare the y-coordinates of \(C\) and \(D\). - Use the x-coordinate to find distance from the y-axis. - For a horizontal segment, subtract the x-coordinates.

Solution

1. Both points have y-coordinate \(8\), so \(h\) is horizontal and is \(8\) units from the x-axis. 2. Point \(C\) has x-coordinate \(2\), so it is \(2\) units from the y-axis. 3. Since \(C\) and \(D\) have the same y-coordinate, \(CD=6-2=4\) units.

Answer

a) \(8\) units b) \(2\) units c) \(4\) units
5188245
Point \(P\) has coordinates \((4, 7)\). Line \(g\) is the horizontal line containing all points with y-coordinate \(2\). a) How far is \(P\) from line \(g\)? b) Point \(Q\) lies on \(g\) directly below \(P\). Give the coordinates of \(Q\).

Hints

- A horizontal line has a constant y-coordinate. - Points directly above or below each other have the same x-coordinate. - Subtract the y-coordinates to find the vertical distance.

Solution

1. Line \(g\) has equation \(y=2\). The vertical distance from \(P\) to \(g\) is \(7-2=5\) units. 2. A point directly below \(P\) has the same x-coordinate, \(4\). Because \(Q\) lies on \(g\), its y-coordinate is \(2\). Thus, \(Q = (4, 2)\).

Answer

a) \(5\) units b) \(Q(4, 2)\)
5188675
Line \(g\) passes through \(A(4, 2)\) and \(B(4, 10)\). Find the distance from \(P(1, 6)\) to line \(g\).

Hints

- Compare the x-coordinates of \(A\) and \(B\). - Decide whether \(g\) is horizontal or vertical. - For a vertical line, compare the point's x-coordinate with the line's x-coordinate.

Solution

1. Points \(A\) and \(B\) have the same x-coordinate, so \(g\) is the vertical line \(x=4\). 2. The horizontal distance from \(P(1, 6)\) to \(g\) is \(4-1=3\) units.

Answer

\(3\) units
5188685
Points \(C(2, 5)\) and \(D(12, 5)\) determine line \(h\). Find the distance from \(Q(7, 11)\) to \(h\).

Hints

- Compare the y-coordinates of \(C\) and \(D\). - Decide whether \(h\) is horizontal or vertical. - For a horizontal line, compare the point's y-coordinate with the line's y-coordinate.

Solution

1. Points \(C\) and \(D\) have the same y-coordinate, so \(h\) is the horizontal line \(y=5\). 2. The vertical distance from \(Q(7, 11)\) to \(h\) is \(11-5=6\) units.

Answer

\(6\) units
5504665
Point \(M\) is in the first quadrant. It is \(6\) units from the y-axis and \(4\) units from the x-axis. What are the coordinates of \(M\)?

Hints

- Connect horizontal distance from the y-axis to one coordinate. - Connect vertical distance from the x-axis to the other coordinate. - Keep the coordinates in their standard order.

Solution

1. A first-quadrant point \(6\) units from the y-axis has x-coordinate \(6\). 2. A point \(4\) units from the x-axis has y-coordinate \(4\). 3. Therefore, \(M=(6, 4)\).

Answer

\(M(6, 4)\)
5504675
Four points are shown on the coordinate plane. a) Which two points lie on the same vertical line? b) Which two points lie on the same horizontal line? c) What coordinate do the points in part a) share? What coordinate do the points in part b) share?
Figure for problem 550467

Hints

- Look for points that line up directly above and below each other. - Look for points that line up directly left and right of each other. - Compare the ordered pairs for each aligned pair.

Solution

1. Points \(A(2, 1)\) and \(B(2, 5)\) have the same x-coordinate, so they lie on the same vertical line. 2. Points \(B(2, 5)\) and \(C(5, 5)\) have the same y-coordinate, so they lie on the same horizontal line. 3. Points \(A\) and \(B\) share x-coordinate \(2\). Points \(B\) and \(C\) share y-coordinate \(5\).

Answer

a) \(A\) and \(B\) b) \(B\) and \(C\) c) They share x-coordinate \(2\) in part a) and y-coordinate \(5\) in part b).
5504685
Point \(P\) is shown on a coordinate plane where the numbered tick marks increase by \(2\). a) What are the coordinates of \(P\)? b) How many small grid spaces are there between \(x=4\) and \(x=6\)? c) Explain why the number of small grid spaces is not the same as the change between the labeled tick values.
Figure for problem 550468

Hints

- Read the numbers printed on the axes before counting spaces. - Notice how many small intervals fit between two neighboring labeled ticks. - Use the scale consistently on both coordinates.

Solution

1. Point \(P\) is at \((6, 4)\). 2. The grid has one-unit subdivisions, so there are \(2\) small grid spaces between \(x=4\) and \(x=6\). 3. The labeled ticks increase by \(2\), while each small grid space represents \(1\) coordinate unit.

Answer

a) \((6, 4)\) b) \(2\) small grid spaces c) The labeled ticks are \(2\) units apart, but each small grid space is \(1\) unit.
5504695
For each point, tell whether it lies on the x-axis, on the y-axis, on both axes, or on neither axis. a) \(A(0, 6)\) b) \(B(4, 0)\) c) \(C(3, 2)\) d) \(D(0, 0)\)

Hints

- Think about what must be true of a point's vertical position to lie on the x-axis. - Think about what must be true of a point's horizontal position to lie on the y-axis. - Treat the origin as a special case where the axes meet.

Solution

1. \(A(0, 6)\) has x-coordinate \(0\), so it lies on the y-axis. 2. \(B(4, 0)\) has y-coordinate \(0\), so it lies on the x-axis. 3. \(C(3, 2)\) has neither coordinate equal to \(0\), so it lies on neither axis. 4. \(D(0, 0)\) is the origin, so it lies on both axes.

Answer

a) y-axis b) x-axis c) neither axis d) both axes
5504705
On a neighborhood map, the first coordinate tells how many blocks east a place is from the starting corner, and the second coordinate tells how many blocks north it is. The community center is at \(C(5, 3)\). a) What does the \(5\) mean? b) What does the \(3\) mean? c) A playground is at \((5, 7)\). How is its east-west position related to the community center's position?

Hints

- Match each coordinate with the direction named in the map description. - Read the ordered pair in its given order. - Compare the first coordinates in part c).

Solution

1. The first coordinate is the east-west position, so \(5\) means the community center is \(5\) blocks east of the starting corner. 2. The second coordinate is the north-south position, so \(3\) means it is \(3\) blocks north. 3. The playground also has first coordinate \(5\), so it is the same distance east as the community center.

Answer

a) \(5\) blocks east b) \(3\) blocks north c) They have the same east-west position.
5188035
Point \(A(7, 5)\) is plotted in the coordinate plane. a) How far is \(A\) from the y-axis? b) How far is \(A\) from the x-axis? c) Point \(B\) lies on the y-axis above the x-axis and is the same distance from the x-axis as \(A\). Give the coordinates of \(B\) and its distance from the origin.

Hints

- The x-coordinate tells how far a first-quadrant point is from the y-axis. - The y-coordinate tells how far it is from the x-axis. - A point on the y-axis has x-coordinate \(0\).

Solution

1. The distance from \(A\) to the y-axis is its x-coordinate, so the distance is \(7\) units. 2. The distance from \(A\) to the x-axis is its y-coordinate, so the distance is \(5\) units. 3. A point on the y-axis has x-coordinate \(0\). Because \(B\) is above the x-axis and \(5\) units from it, \(B = (0, 5)\). 4. Point \(B\) is \(5\) units from the origin.

Answer

a) \(7\) units b) \(5\) units c) \(B(0, 5)\); \(5\) units from the origin
5188045
Point \(S\) lies in Quadrant I. It is \(4\,\text{cm}\) from the y-axis. Its distance from the x-axis is \(3\,\text{cm}\) greater than its distance from the y-axis. One coordinate unit represents \(1\,\text{cm}\). a) Give the coordinates of \(S\). b) Line \(g\) is parallel to the x-axis and passes through \((0,12)\). How far is \(S\) from line \(g\)?

Hints

- Use the distance from each axis to determine the corresponding coordinate. - First find the distance from the x-axis. - A line parallel to the x-axis has a constant y-coordinate.

Solution

1. In Quadrant I, the x-coordinate equals the distance from the y-axis, so \(x=4\). 2. The y-coordinate is \(4+3=7\), so \(S=(4,7)\). 3. Line \(g\) is the horizontal line \(y=12\). The distance from \(S\) to \(g\) is \(12-7=5\,\text{cm}\).

Answer

a) \(S(4, 7)\) b) \(5\,\text{cm}\)
5188255
Three straight park paths are vertical. Every point on \(w_1\) has x-coordinate \(3\), every point on \(w_2\) has x-coordinate \(8\), and every point on \(w_3\) has x-coordinate \(15\). Leo chooses \(A(3, 2)\) on \(w_1\) and \(B(8, 9)\) on \(w_2\). He says the paths are \(7\) units apart because \(9-2=7\). a) Explain Leo's error. b) Find the distance between \(w_1\) and \(w_2\). c) Find the distance between \(w_2\) and \(w_3\), and between \(w_1\) and \(w_3\).

Hints

- Decide whether the distance between vertical paths is horizontal or vertical. - Ask which coordinate stays fixed along each vertical path. - The locations of \(A\) and \(B\) along their paths should not change the distance between the paths.

Solution

1. The paths are vertical, so their separation is horizontal. The y-coordinates of two chosen points do not determine the distance between the paths. 2. For \(w_1\) and \(w_2\), compare their fixed x-coordinates: \(8-3=5\) units. 3. For \(w_2\) and \(w_3\), \(15-8=7\) units. 4. For \(w_1\) and \(w_3\), \(15-3=12\) units.

Answer

a) Leo compared y-coordinates, but the distance between vertical paths is determined by their x-coordinates. b) \(5\) units c) \(7\) units; \(12\) units
5188435
Line \(h\) passes through \(P(2, 5)\) and \(Q(2, 10)\). a) Find the length of \(\overline{PQ}\). b) Point \(R\) also lies on \(h\) and is exactly \(3\) units from \(P\). Give all possible coordinates of \(R\) for which both coordinates are positive.

Hints

- Every point on a vertical line has the same x-coordinate. - Move the given distance in both directions along the line. - Subtract the y-coordinates to find a vertical distance.

Solution

1. Points \(P\) and \(Q\) have the same x-coordinate, so \(\overline{PQ}\) is vertical. Its length is \(10-5=5\) units. 2. Every point on \(h\) has x-coordinate \(2\). 3. A point \(3\) units from \(P(2, 5)\) on this vertical line can have y-coordinate \(5+3=8\) or \(5-3=2\). Therefore, the possibilities are \(R_1(2, 8)\) and \(R_2(2, 2)\).

Answer

a) \(5\) units b) \(R(2, 8)\) or \(R(2, 2)\)
5504715
Sam was asked to plot the point \(S(2, 5)\). The graph shows where Sam placed \(S\). a) What coordinates did Sam actually plot? b) What mistake did Sam most likely make? c) Where should \(S\) be plotted instead?
Figure for problem 550471

Hints

- Read the plotted point from the axes before comparing it with the requested point. - Compare the first coordinate in the request with the horizontal position shown. - Decide whether the two requested coordinates were used in the wrong order.

Solution

1. The plotted point is at \((5, 2)\). 2. Sam most likely reversed the x-coordinate and y-coordinate. 3. The correct location is \((2, 5)\).

Answer

a) \((5, 2)\) b) Sam reversed the coordinates. c) \((2, 5)\)
5504725
Point \(Q\) is at \((2, 5)\). Point \(P\) is in the first quadrant, is \(7\) units from the y-axis, and lies on the same horizontal line as \(Q\). Point \(R\) lies on the same vertical line as \(P\), and its y-coordinate is \(3\) greater than the y-coordinate of \(Q\). Find the coordinates of \(P\) and \(R\).

Hints

- Use each condition to determine one coordinate at a time. - The first-quadrant condition tells you which side of the y-axis \(P\) is on. - Same horizontal line and same vertical line preserve different coordinates.

Solution

1. Because \(P\) is in the first quadrant and is \(7\) units from the y-axis, its x-coordinate is \(7\). 2. Because \(P\) is on the same horizontal line as \(Q(2, 5)\), its y-coordinate is \(5\). Thus, \(P=(7, 5)\). 3. Point \(R\) has the same x-coordinate as \(P\), so its x-coordinate is \(7\). 4. The y-coordinate of \(R\) is \(5+3=8\). Thus, \(R=(7, 8)\).

Answer

\(P(7, 5)\) and \(R(7, 8)\)
5504735
Points \(A\), \(B\), and \(C\) are shown on the coordinate plane. a) Find the horizontal distance from \(A\) to \(B\). b) Find the vertical distance from \(B\) to \(C\). c) Which coordinate stays the same from \(A\) to \(B\)? Which coordinate stays the same from \(B\) to \(C\)?
Figure for problem 550473

Hints

- First decide whether each segment is horizontal or vertical. - For a horizontal segment, compare horizontal positions. - For a vertical segment, compare vertical positions.

Solution

1. Points \(A(1, 2)\) and \(B(6, 2)\) have the same y-coordinate, so the horizontal distance is \(6-1=5\) units. 2. Points \(B(6, 2)\) and \(C(6, 7)\) have the same x-coordinate, so the vertical distance is \(7-2=5\) units. 3. The y-coordinate stays the same from \(A\) to \(B\). The x-coordinate stays the same from \(B\) to \(C\).

Answer

a) \(5\) units b) \(5\) units c) The y-coordinate stays the same from \(A\) to \(B\), and the x-coordinate stays the same from \(B\) to \(C\).
5504745
Point \(P\) lies above the x-axis and is \(4\) units from it. Its x-coordinate is greater than \(5\) and less than \(8\), and it is even. What are the coordinates of \(P\)? Explain how the conditions determine one point.

Hints

- Use the distance from the x-axis and the word “above” to determine the y-coordinate. - Focus on the numbers strictly between \(5\) and \(8\). - Use the parity condition to select the x-coordinate.

Solution

1. Because \(P\) is above the x-axis and \(4\) units from it, its y-coordinate is \(4\). 2. The only even number greater than \(5\) and less than \(8\) is \(6\), so the x-coordinate is \(6\). 3. Therefore, \(P=(6, 4)\).

Answer

\(P(6, 4)\)

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