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Coordinate plane in four quadrants

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5349306
Points \(D\), \(E\), and \(F\) are shown on the coordinate plane. Read the graph to determine their coordinates.
Figure for problem 534930

Hints

- From the origin, read the horizontal coordinate first and the vertical coordinate second. - A point on an axis has one coordinate equal to \(0\).

Solution

1. Point \(D\) is on the x-axis at \(x=-2\), so \(D=(-2, 0)\). 2. Point \(E\) is on the y-axis at \(y=-1\), so \(E=(0, -1)\). 3. Point \(F\) is on the x-axis at \(x=2\), so \(F=(2, 0)\).

Answer

\(D=(-2, 0)\), \(E=(0, -1)\), and \(F=(2, 0)\)
5542606
To plot point \(T\), start at the origin, move \(4\) units left, and then move \(3\) units up. a) What ordered pair should be plotted? b) In which quadrant does \(T\) lie?

Hints

- Horizontal movement determines the x-coordinate. - Vertical movement determines the y-coordinate. - Use the signs of the two coordinates to identify the quadrant.

Solution

1. Moving \(4\) units left gives x-coordinate \(-4\). 2. Moving \(3\) units up gives y-coordinate \(3\). 3. Therefore, \(T=(-4, 3)\). A point with negative x-coordinate and positive y-coordinate lies in Quadrant II.

Answer

a) \((-4, 3)\) b) Quadrant II
5187306
Point \(A(15, 22)\) lies on a line parallel to the x-axis. Give the coordinates of three different points that also lie on this line.

Hints

- What coordinate is shared by all points on a horizontal line? - Which coordinate changes as you move parallel to the x-axis? - Choose three different values for the x-coordinate.

Solution

1. Every point on a horizontal line has the same y-coordinate. 2. Since \(A\) has y-coordinate \(22\), every point on the line has the form \((x, 22)\). 3. Choose three distinct x-coordinates other than \(15\). For example, \((0, 22)\), \((10, 22)\), and \((30, 22)\) all lie on the line.

Answer

One possible answer is \((0, 22)\), \((10, 22)\), and \((30, 22)\). Any three distinct points of the form \((x, 22)\), different from \(A\), are correct.
5187316
Point \(B(-10, 35)\) lies on line \(h\), which is parallel to the y-axis. Which of the following points also lie on \(h\)? \(P_1(-10, 0)\), \(P_2(10, 35)\), \(P_3(-10, -5)\), \(P_4(0, 35)\), \(P_5(-10, 100)\)

Hints

- Which coordinate stays constant on a vertical line? - Compare the first coordinate of \(B\) with the first coordinate of each listed point. - The y-coordinate can vary on a line parallel to the y-axis.

Solution

1. Every point on a vertical line has the same x-coordinate. 2. Because \(B\) has x-coordinate \(-10\), every point on \(h\) must also have x-coordinate \(-10\). 3. The points with x-coordinate \(-10\) are \(P_1\), \(P_3\), and \(P_5\).

Answer

\(P_1(-10, 0)\), \(P_3(-10, -5)\), and \(P_5(-10, 100)\) lie on line \(h\).
5190766
A world map uses a coordinate system based on longitude and latitude. Longitude is the x-coordinate: east is positive and west is negative. Latitude is the y-coordinate: north is positive and south is negative. The table uses rounded coordinates. | Place | Coordinates \((x, y)\) | | :--- | :--- | | Cairo | \((31, 30)\) | | London | \((0, 51)\) | | Quito | \((-78, 0)\) | | Melbourne | \((145, -38)\) | a) Which place lies on the x-axis, which represents the equator? b) Which place is farthest south? c) Which place is farthest east? d) Which place lies on the y-axis, which represents the prime meridian?

Hints

- Points on the x-axis have y-coordinate \(0\). - Farther south means a smaller y-coordinate. - Farther east means a larger x-coordinate. - Points on the y-axis have x-coordinate \(0\).

Solution

1. A point on the x-axis has y-coordinate \(0\). Quito has coordinates \((-78, 0)\), so Quito lies on the equator. 2. The place farthest south has the least y-coordinate. Since \(-38\) is the least y-coordinate, Melbourne is farthest south. 3. The place farthest east has the greatest x-coordinate. Since \(145\) is the greatest x-coordinate, Melbourne is farthest east. 4. A point on the y-axis has x-coordinate \(0\). London has coordinates \((0, 51)\), so London lies on the prime meridian.

Answer

a) Quito b) Melbourne c) Melbourne d) London
5331436
Find the coordinates of points \(A\) through \(F\) in the graph. For each point, state its quadrant or name the coordinate axis on which it lies.
Figure for problem 533143

Hints

- Read the x-coordinate first and the y-coordinate second. - Quadrants are numbered counterclockwise beginning in the upper-right region. - A point lies on an axis when one coordinate is \(0\).

Solution

1. Reading the graph gives \(A(4, 3)\), \(B(-3, 2)\), \(C(-2, -4)\), \(D(3, -1)\), \(E(0, 2)\), and \(F(-5, 0)\). 2. Point \(A\) is in Quadrant I, \(B\) is in Quadrant II, \(C\) is in Quadrant III, and \(D\) is in Quadrant IV. 3. Point \(E\) lies on the y-axis because its x-coordinate is \(0\). Point \(F\) lies on the x-axis because its y-coordinate is \(0\).

Answer

\(A(4, 3)\), Quadrant I; \(B(-3, 2)\), Quadrant II; \(C(-2, -4)\), Quadrant III; \(D(3, -1)\), Quadrant IV; \(E(0, 2)\), y-axis; \(F(-5, 0)\), x-axis.
5331456
The graph shows points \(A\), \(B\), \(C\), \(D\), and \(E\) connected in alphabetical order, with \(E\) connected back to \(A\). a) What geometric name describes the closed outline? b) Which two sides are vertical, and what is the length of each vertical side? c) What are the coordinates of the highest vertex?
Figure for problem 533145

Hints

- Count the sides of the closed outline. - A vertical side has endpoints with the same x-coordinate; use the y-coordinates to find its length. - The highest vertex has the greatest y-coordinate.

Solution

1. The closed outline has five sides, so it is a pentagon. 2. Sides \(BC\) and \(EA\) are vertical because each pair of endpoints has the same x-coordinate. Each extends from \(y=-2\) to \(y=1\), so each has length \(3\) units. 3. The highest vertex is \(D=(0, 3)\).

Answer

a) Pentagon b) \(BC\) and \(EA\); each is \(3\) units long. c) \((0, 3)\)
5331466
The graph shows ten labeled points connected by segments. a) Which labeled points lie on either coordinate axis? b) Write the coordinates of points \(3\), \(6\), and \(9\).
Figure for problem 533146

Hints

- Points on the x-axis have y-coordinate \(0\); points on the y-axis have x-coordinate \(0\). - Use the labels to locate the requested points. - Read the x-coordinate before the y-coordinate.

Solution

1. A point lies on the y-axis when its x-coordinate is \(0\), and it lies on the x-axis when its y-coordinate is \(0\). 2. From the graph, points \(1\) and \(6\) lie on the y-axis, while points \(4\) and \(8\) lie on the x-axis. 3. Reading the requested coordinates gives point \(3=(4,2)\), point \(6=(0,-1)\), and point \(9=(-4,2)\).

Answer

a) Points \(1,4,6,8\) b) Point \(3\) is \((4,2)\), point \(6\) is \((0,-1)\), and point \(9\) is \((-4,2)\).
5331726
A triangle with vertices \(A\), \(B\), and \(C\) is shown on the coordinate plane. a) Find the coordinates of all three vertices. b) Which point has a negative x-coordinate and a positive y-coordinate?
Figure for problem 533172

Hints

- Read the x-coordinate first and the y-coordinate second. - Coordinates left of the y-axis have negative x-values. - Coordinates above the x-axis have positive y-values.

Solution

1. Reading the graph gives \(A(4, 1)\), \(B(-3, 4)\), and \(C(-2, -5)\). 2. Point \(B\) has a negative x-coordinate and a positive y-coordinate, so it lies in Quadrant II.

Answer

a) \(A(4, 1)\), \(B(-3, 4)\), and \(C(-2, -5)\) b) Point \(B\)
5331736
Points \(A\), \(B\), \(C\), and \(D\), along with line \(g\), are shown on the coordinate plane. a) Find the coordinates of all four points. b) Which points lie on line \(g\)?
Figure for problem 533173

Hints

- Read the x-coordinate first and the y-coordinate second. - Trace line \(g\) across the grid. - Check whether the center of each plotted point lies on the displayed line.

Solution

1. Reading the graph gives \(A=(-4, 5)\), \(B=(2, 2)\), \(C=(5, -3)\), and \(D=(-3, -3)\). 2. Inspecting the plotted line shows that points \(B\) and \(D\) lie directly on line \(g\). Points \(A\) and \(C\) do not.

Answer

a) \(A=(-4, 5)\), \(B=(2, 2)\), \(C=(5, -3)\), and \(D=(-3, -3)\) b) Points \(B\) and \(D\)
5349126
The vertices of a simple house outline are marked on the coordinate plane. Find the coordinates of points \(P\), \(Q\), \(R\), \(S\), and \(T\).
Figure for problem 534912

Hints

- Read the x-coordinate first. - Then read the y-coordinate. - A point on an axis has one coordinate equal to \(0\).

Solution

1. Reading the graph gives \(P(-2, 0)\), \(Q(2, 0)\), \(R(2, 3)\), \(S(0, 5)\), and \(T(-2, 3)\).

Answer

\(P(-2, 0)\), \(Q(2, 0)\), \(R(2, 3)\), \(S(0, 5)\), and \(T(-2, 3)\)
5542616
A point should be plotted at \((-3,2)\). Four candidate points are labeled in Quadrant II. Which labeled point is in the correct location? Explain how both distances from the axes identify it.
Figure for problem 554261

Hints

- All four candidates have the same sign pattern, so quadrant alone is not enough. - Count horizontal distance from the y-axis for the x-coordinate. - Count vertical distance from the x-axis for the y-coordinate.

Solution

1. The x-coordinate \(-3\) means the point must be \(3\) units left of the y-axis. 2. The y-coordinate \(2\) means the point must be \(2\) units above the x-axis. 3. Among the four Quadrant-II candidates, only point \(B\) is exactly \(3\) units left and \(2\) units up.

Answer

Point \(B\)
5187326
Points \(R(7, 12)\) and \(S(7, 40)\) lie on the same line \(g\). a) Is line \(g\) parallel to the x-axis or the y-axis? Explain. b) Point \(T\) lies on \(g\), and its y-coordinate is halfway between the y-coordinates of \(R\) and \(S\). Find the coordinates of \(T\).

Hints

- Compare the two coordinates of \(R\) and \(S\). Which coordinate is the same? - A constant x-coordinate describes what kind of line? - Average the two y-coordinates to find the value halfway between them.

Solution

1. Points \(R\) and \(S\) have the same x-coordinate, \(7\), so the line through them is vertical and parallel to the y-axis. 2. The y-coordinate halfway between \(12\) and \(40\) is \(\frac{12 + 40}{2} = 26\). 3. Every point on \(g\) has x-coordinate \(7\), so \(T = (7, 26)\).

Answer

a) Line \(g\) is parallel to the y-axis because its points have the same x-coordinate. b) \(T = (7, 26)\)
51889010
Line \(g\) passes through \(P=(3, 2)\) and \(Q=(3, 8)\). Line \(h\) passes through \(R=(7, 2)\) and \(S=(7, 8)\). One coordinate unit represents \(1\,\text{cm}\). a) Describe the location of all points that are the same distance from \(g\) and \(h\). b) Describe the location of all points that are \(2\,\text{cm}\) from \(g\). Give the equations of the lines.

Hints

- Identify the equations and directions of \(g\) and \(h\). - Find the x-coordinate halfway between the two parallel lines. - A fixed positive perpendicular distance from a line produces two parallel loci, one on each side.

Solution

1. Line \(g\) is \(x=3\), and line \(h\) is \(x=7\). Both are vertical and parallel. 2. Points equidistant from two parallel lines lie on the parallel line halfway between them. The halfway x-coordinate is \((3+7)\div2=5\), so the locus is \(x=5\). 3. Points \(2\,\text{cm}\) from \(g\) lie on two vertical lines. Their x-coordinates are \(3-2=1\) and \(3+2=5\), so the lines are \(x=1\) and \(x=5\).

Answer

a) The points lie on the vertical line \(x=5\), halfway between \(g\) and \(h\). b) The points lie on the vertical lines \(x=1\) and \(x=5\).
5411586
Reflect \(E\) across the y-axis and then across the x-axis. a) Give the coordinates after the first reflection. b) Give the final coordinates. c) Name the final quadrant.
Figure for problem 541158

Hints

- Read the original ordered pair from the coordinate plane. - Apply the two reflections one at a time. - Match the final coordinate signs to a quadrant.

Solution

1. The graph shows \(E=(-7, 2)\). 2. Reflecting \((-7, 2)\) across the y-axis gives \((7, 2)\). 3. Reflecting \((7, 2)\) across the x-axis gives \((7, -2)\). 4. A positive x-coordinate and a negative y-coordinate place the final point in Quadrant IV.

Answer

a) \((7, 2)\) b) \((7, -2)\) c) Quadrant IV
5411596
Point \(T\) lies in Quadrant III. It is \(4\) units from the y-axis and \(6\) units from the x-axis. Find the coordinates of \(T\).

Hints

- Relate distance from each axis to one coordinate. - Determine the possible magnitudes before choosing signs. - Use the quadrant to select the correct sign for each coordinate.

Solution

1. A distance of \(4\) units from the y-axis means \(|x|=4\). 2. A distance of \(6\) units from the x-axis means \(|y|=6\). 3. In Quadrant III, both coordinates are negative, so \(T=(-4, -6)\).

Answer

\((-4, -6)\)
5411606
a) Across which axis is \(A\) reflected to reach \(B\)? b) Across which axis is \(B\) reflected to reach \(C\)? c) Which coordinate signs change from \(A\) to \(C\)?
Figure for problem 541160

Hints

- Read each ordered pair from the coordinate plane. - Compare one coordinate at a time for each pair of points. - A reflection keeps the coordinate along the mirror axis unchanged.

Solution

1. The graph shows \(A=(5, 4)\), \(B=(-5, 4)\), and \(C=(-5, -4)\). 2. From \(A\) to \(B\), only the x-coordinate changes sign, so the reflection is across the y-axis. 3. From \(B\) to \(C\), only the y-coordinate changes sign, so the reflection is across the x-axis. 4. From \(A\) to \(C\), both coordinate signs change.

Answer

a) The y-axis b) The x-axis c) Both the x-coordinate sign and the y-coordinate sign
5411616
Find the length of Route 1 and Route 2 from \(P\) to \(Q\). Explain why the lengths are equal.
Figure for problem 541161

Hints

- Read the endpoints and corner points from the graph. - Split the horizontal distance at the y-axis instead of subtracting a negative coordinate. - Compare the horizontal and vertical lengths used by the two routes.

Solution

1. The graph shows \(P=(-2, 1)\) and \(Q=(6, 5)\). 2. Horizontally, there are \(2\) units from \(x=-2\) to the y-axis and \(6\) more units from the y-axis to \(x=6\), for \(8\) units total. Vertically, there are \(5-1=4\) units. 3. Route 1 has length \(8+4=12\) units. Route 2 has length \(4+8=12\) units. 4. The routes use the same horizontal and vertical distances in opposite order, so their total lengths are equal.

Answer

Route 1 is \(12\) units, and Route 2 is \(12\) units. They are equal because both use \(8\) horizontal units and \(4\) vertical units.
5411626
Reflect \(L\) across the y-axis. a) Find the image point. b) Does the image lie in a quadrant? Explain. c) What is the distance between the original point and its image?
Figure for problem 541162

Hints

- Read the original ordered pair from the coordinate plane. - Reflect the horizontal position across the y-axis. - Compare each point's horizontal distance from the y-axis.

Solution

1. The graph shows \(L=(8, 0)\). 2. Reflection across the y-axis changes the sign of the x-coordinate, giving \((-8, 0)\). 3. A point with y-coordinate \(0\) lies on the x-axis, not in a quadrant. 4. The original point is \(8\) units to the right of the y-axis and its image is \(8\) units to the left, so the points are \(8+8=16\) units apart.

Answer

a) \((-8, 0)\) b) No; it lies on the x-axis. c) \(16\) units
5411656
a) Find the point where segment \(\overline{RS}\) crosses the y-axis. b) Find the distance from \(R\) to that crossing point. c) Find the distance from the crossing point to \(S\).
Figure for problem 541165

Hints

- Read the endpoint coordinates from the graph. - A point on the y-axis has x-coordinate \(0\). - Use each endpoint's horizontal distance from the y-axis.

Solution

1. The graph shows \(R=(-7, 4)\) and \(S=(5, 4)\). 2. A point on the y-axis has x-coordinate \(0\), and the segment has y-coordinate \(4\), so the crossing point is \((0, 4)\). 3. Point \(R\) is \(7\) horizontal units from the y-axis, so the distance from \(R\) to the crossing point is \(7\) units. 4. Point \(S\) is \(5\) horizontal units from the y-axis, so the distance from the crossing point to \(S\) is \(5\) units.

Answer

a) \((0, 4)\) b) \(7\) units c) \(5\) units
5411666
A student says, “The y-coordinates are \(7\) and \(-8\), so the distance is \(7-8=-1\) unit.” Explain two errors in this reasoning and find the correct distance between \(K\) and \(L\).
Figure for problem 541166

Hints

- Notice where each point lies relative to the x-axis. - Think about how far each point is from the x-axis and how those two distances combine. - Check whether a distance can ever be negative.

Solution

1. The graph shows \(K=(-2, 7)\) and \(L=(-2, -8)\). 2. The first error is subtracting \(8\) from \(7\). The points are on opposite sides of the x-axis, so their vertical distances from the x-axis must be combined. 3. Point \(K\) is \(7\) units above the x-axis and point \(L\) is \(8\) units below it. Their distance is \(7+8=15\) units. 4. The second error is reporting a negative distance. Distance is a nonnegative length.

Answer

The student subtracted the two distances from the x-axis instead of combining them, and a distance cannot be negative. The correct distance is \(15\) units.
5411676
On the park map, a ranger starts at \(A\), moves horizontally to \(x=6\), and then moves straight down \(9\) units. a) Give the coordinates where the ranger turns. b) Give the final coordinates. c) Find the total distance traveled.
Figure for problem 541167

Hints

- Read the starting point from the coordinate grid. - For a move that crosses an axis, split the distance at the axis. - Add the horizontal and vertical segment lengths to find the route distance.

Solution

1. The graph shows \(A=(-3, 4)\). Moving horizontally to \(x=6\) keeps the y-coordinate \(4\), so the turn is at \((6, 4)\). 2. From y-coordinate \(4\), moving down \(4\) units reaches the x-axis and moving \(5\) more units completes the \(9\)-unit move. The final point is \((6, -5)\). 3. Horizontally, there are \(3\) units from \(x=-3\) to the y-axis and \(6\) more units to \(x=6\), for \(9\) units total. 4. The route length is \(9+9=18\) units.

Answer

a) \((6, 4)\) b) \((6, -5)\) c) \(18\) units
5411686
a) Find the fourth vertex of the rectangle. b) List the quadrants containing the four vertices. c) Does any side cross the x-axis? Does any side cross the y-axis?
Figure for problem 541168

Hints

- Read the three known coordinates from the graph. - An axis-aligned rectangle uses two x-values and two y-values in all combinations. - Check whether each side connects coordinates with opposite signs.

Solution

1. The graph shows the known vertices \((-2, 5)\), \((7, 5)\), and \((7, -4)\). 2. The missing vertex combines x-coordinate \(-2\) with y-coordinate \(-4\), giving \((-2, -4)\). 3. The four vertices lie in Quadrants II, I, IV, and III, so all four quadrants are represented. 4. The vertical sides connect positive and negative y-values, so they cross the x-axis. The horizontal sides connect negative and positive x-values, so they cross the y-axis.

Answer

a) \((-2, -4)\) b) Quadrants I, II, III, and IV c) Yes to both. The vertical sides cross the x-axis, and the horizontal sides cross the y-axis.
5411636
Point \(W\) has the same x-coordinate as \(N\) and is \(9\) units from \(N\). No direction is specified. a) Find both possible coordinates for \(W\). b) Name the quadrant containing each possible point.
Figure for problem 541163

Hints

- Read the coordinate of \(N\) from the graph. - A vertical distance can be reached in two directions. - When moving downward past zero, split the \(9\)-unit move at the x-axis.

Solution

1. The graph shows \(N=(2, 2)\), and the x-coordinate of \(W\) must remain \(2\). 2. Moving \(9\) units up from y-coordinate \(2\) reaches \(11\), so one possible point is \((2, 11)\). 3. Moving down, there are \(2\) units from \(2\) to \(0\) and \(7\) more units to complete a \(9\)-unit move, so the other y-coordinate is \(-7\). The other point is \((2, -7)\). 4. The first point is in Quadrant I, and the second is in Quadrant IV.

Answer

a) \((2, 11)\) and \((2, -7)\) b) \((2, 11)\) is in Quadrant I; \((2, -7)\) is in Quadrant IV.
5411646
Point \(Q\) has the same x-coordinate as \(P\), is \(14\) units from \(P\), and segment \(PQ\) crosses the x-axis. Find \(Q\).
Figure for problem 541164

Hints

- Read \(P\) from the graph and keep its x-coordinate fixed. - Use part of the \(14\)-unit distance to reach the x-axis first. - The crossing condition tells you which vertical direction is possible.

Solution

1. The graph shows \(P=(-4, -6)\), so \(Q\) must also have x-coordinate \(-4\). 2. To move \(14\) units upward from y-coordinate \(-6\), use \(6\) units to reach the x-axis and \(8\) more units above it. This gives y-coordinate \(8\). 3. Moving \(14\) units downward would keep both points below the x-axis, so that segment would not cross the x-axis. 4. Therefore, \(Q=(-4, 8)\).

Answer

\((-4, 8)\)

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