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Understand the coordinate plane

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5190955
A world map uses longitude as the x-coordinate and latitude as the y-coordinate. A ship is located exactly where the equator crosses the prime meridian. What ordered pair gives the ship’s location?

Hints

- The first coordinate represents longitude. - The second coordinate represents latitude. - Both reference lines have value \(0^\circ\).

Solution

1. The equator has latitude \(0^\circ\), so the y-coordinate is \(0\). 2. The prime meridian has longitude \(0^\circ\), so the x-coordinate is \(0\). 3. Therefore, the ship’s location is \((0, 0)\).

Answer

\((0, 0)\)
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) \(Q(4, 2)\)
5188255
Three straight, parallel paths in a park are represented by the vertical lines \(w_1: x=3\), \(w_2: x=8\), and \(w_3: x=15\). Find the distance between \(w_1\) and \(w_2\), the distance between \(w_2\) and \(w_3\), and the distance between the two outer paths \(w_1\) and \(w_3\).

Hints

- Vertical lines have constant x-coordinates. - Subtract the x-coordinates to find the distance between two vertical lines. - The two smaller distances combine to give the outer distance.

Solution

1. The distance between \(w_1\) and \(w_2\) is \(8-3=5\) units. 2. The distance between \(w_2\) and \(w_3\) is \(15-8=7\) units. 3. The distance between \(w_1\) and \(w_3\) is \(15-3=12\) units. This also equals \(5+7=12\).

Answer

\(w_1\) to \(w_2\): \(5\) units \(w_2\) to \(w_3\): \(7\) units \(w_1\) to \(w_3\): \(12\) units
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, use the difference between x-coordinates.

Solution

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

Solution

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

Answer

\(6\) units
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) \(5\,\text{cm}\)
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 \(R(2, 2)\)

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