📝 Rectangular coordinate system explained (16 MCQs)

📖 From Digital SAT Algebra • 4. Graphs • 16 questions available

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What is Rectangular coordinate system explained?

Definition:
The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane formed by two perpendicular number lines called axes—the horizontal x-axis and the vertical y-axis—that intersect at the origin (0,0), dividing the plane into four quadrants, and it provides a way to locate any point using an ordered pair (x, y), representing its horizontal and vertical distances from the origin, making it fundamental for graphing equations and analyzing relationships between variables.

Working:
This system works by using the x-coordinate (abscissa) to indicate the position left or right of the origin, and the y-coordinate (ordinate) to indicate the position up or down; the x-coordinate is positive to the right and negative to the left, while the y-coordinate is positive upward and negative downward; every point on the plane corresponds to a unique ordered pair, and the axes are scaled to accommodate the range of values, and the distance between two points can be calculated using the distance formula derived from the Pythagorean theorem $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

Example:
A simple example is plotting the point (3, 2): from the origin, move 3 units to the right along the x-axis, then 2 units up parallel to the y-axis, and mark the point; another example is the point (-2, -4), which is 2 units left and 4 units down, illustrating how the coordinate system locates points in all four quadrants.

Reason:
The rectangular coordinate system is essential in algebra and science because it allows us to visualize mathematical relationships, graph functions, solve systems of equations, and model real-world phenomena, making it a foundational tool in mathematics, physics, engineering, and data analysis.

Easy

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📝 All Rectangular coordinate system explained MCQs

Q1. A point has coordinates $(-4,5)$ . Without plotting it, which statement correctly describes its location and the signs of its coordinates?

A.It lies in Quadrant II, where

x<0

and

y>0

. ✅

B.It lies in Quadrant I, where

x>0

and

y>0

C.It lies in Quadrant III, where

x<0

and

y<0

D.It lies in Quadrant IV, where

x>0

and

y<0

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The first coordinate gives the horizontal position and the second gives the vertical position. Since $x=-4$ is negative and $y=5$ is positive, the point lies left of the vertical axis and above the horizontal axis, which is Quadrant II.

Q2. A point moves from $(2,-3)$ to $(2,4)$ . What can be concluded about the movement without calculating a distance?

A.It moves 7 units horizontally.

B.It moves 7 units vertically while its horizontal position remains unchanged. ✅

C.It moves 6 units vertically while its horizontal position remains unchanged.

D.It moves diagonally because both coordinates must change.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The $x$ -coordinate remains $2$ , so there is no horizontal movement. The $y$ -coordinate changes from $-3$ to $4$ , giving a vertical change of $7$ units. Therefore, the movement is entirely vertical.

Q3. A student claims that $(-6,-2)$ is in Quadrant II because the first coordinate is negative. Which reasoning best evaluates the claim?

A.Correct, because only the sign of

x

determines the quadrant.

B.Correct, because negative coordinates always indicate Quadrant II.

C.Incorrect, because both coordinates are negative, placing the point in Quadrant III. ✅

D.Incorrect, because a point with two negative coordinates lies on an axis.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A quadrant is determined by the signs of both coordinates. For $(-6,-2)$ , both $x$ and $y$ are negative, so the point is left of the vertical axis and below the horizontal axis, placing it in Quadrant III.

Q4. Two locations are represented by $A=(-3,4)$ and $B=(5,4)$ . Which interpretation is most useful when modeling the change from $A$ to $B$ ?

A.The movement is 8 units horizontally with no vertical change. ✅

B.The movement is 8 units vertically with no horizontal change.

C.The movement is 2 units diagonally because the coordinates have different signs.

D.The movement is 4 units horizontally and 5 units vertically.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Both points have the same $y$ -coordinate, so they lie on the same horizontal level. The $x$ -coordinate increases from $-3$ to $5$ , a change of $8$ , while the vertical coordinate remains unchanged.

Q5. A rectangular map uses coordinates to locate objects. A sensor is moved from $(-2,-1)$ to $(3,6)$ . Which sequence correctly describes its horizontal and vertical coordinate changes?

A.5 units left and 7 units down

B.5 units right and 7 units up ✅

C.1 unit right and 5 units up

D.7 units right and 5 units up

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The horizontal change is $3-(-2)=5$ , so the sensor moves 5 units right. The vertical change is $6-(-1)=7$ , so it moves 7 units up. Separating the coordinate changes prevents confusion caused by negative values.

Q6. A student says that the points $(4,7)$ and $(7,4)$ represent the same location because they contain the same two numbers. What is the strongest response?

A.The student is correct because coordinate order does not matter.

B.The student is incorrect because the first coordinate controls vertical position and the second controls horizontal position.

C.The student is incorrect because coordinate order determines horizontal and vertical positions, so the locations generally differ. ✅

D.The student is correct only when both coordinates are positive.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Coordinates are ordered pairs, so their order is essential. In $(4,7)$ , the horizontal position is $4$ and the vertical position is $7$ ; in $(7,4)$ , these positions are reversed, producing different locations.

Q7. A delivery robot starts at $(6,2)$ , moves 9 units left, and then 5 units down. Which coordinate represents its final position?

(15,7)

(-3,7)

(-3,-3)

✅

(3,-3)

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Moving 9 units left changes the $x$ -coordinate from $6$ to $6-9=-3$ . Moving 5 units down changes the $y$ -coordinate from $2$ to $2-5=-3$ . Thus the final position is $(-3,-3)$ .

Q8. A school places a fountain at $(-5,3)$ and a gate at $(4,3)$ . A planner wants a straight horizontal path connecting them. Which statement justifies this design?

A.Their

x

-coordinates are equal.

B.Their

y

-coordinates are equal, so both locations lie on the same horizontal level. ✅

C.Their coordinates have opposite signs, so they must form a horizontal line.

D.Their distances from the origin are equal.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Points with equal $y$ -coordinates lie on the same horizontal line. Since both locations have $y=3$ , the path can be horizontal, extending from $x=-5$ through $x=4$ without changing vertical position.

Q9. A graph shows three points $P=(-4,1)$ , $Q=(2,1)$ , and $R=(2,-5)$ . Which comparison correctly describes the segments $PQ$ and $QR$ ?

A.Both are vertical segments of equal length.

PQ

is horizontal and 6 units long, while

QR

is vertical and 6 units long. ✅

PQ

is vertical and 6 units long, while

QR

is horizontal and 6 units long.

D.Both are diagonal segments because their endpoints have different coordinates.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For $PQ$ , the $y$ -coordinates are both $1$ , so the segment is horizontal, and the horizontal change is $2-(-4)=6$ . For $QR$ , the $x$ -coordinates are both $2$ , making it vertical with length $1-(-5)=6$ .

Q10. A plotted point lies 3 units to the left of the vertical axis and 8 units above the horizontal axis. Which ordered pair correctly models its position?

(8,-3)

(-3,8)

✅

(3,8)

(-8,3)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Being 3 units left of the vertical axis means the horizontal coordinate is $-3$ . Being 8 units above the horizontal axis means the vertical coordinate is $8$ . Therefore, the correct ordered pair is $(-3,8)$ .

Q11. A student plots $(5,-4)$ by moving 4 units right and then 5 units down from the origin. What error did the student make?

A.The student reversed the signs of both coordinates.

B.The student used the absolute values correctly but reversed the coordinate roles.

C.The student moved the correct horizontal distance but should have moved 5 units right and 4 units down. ✅

D.The student should have moved 4 units left and 5 units up.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The ordered pair $(5,-4)$ requires moving 5 units right because $x=5$ , followed by 4 units down because $y=-4$ . The student's movements correspond to $(4,-5)$ , so the magnitudes were assigned to the wrong coordinates.

Q12. A map uses the origin as a reference point. Location $A$ is $(-7,2)$ , while location $B$ is $(-7,-6)$ . A planner concludes that $A$ and $B$ are horizontally aligned. How should this conclusion be corrected?

A.They are vertically aligned because their

x

-coordinates are equal. ✅

B.They are horizontally aligned because their

y

-coordinates have opposite signs.

C.They are diagonally aligned because both coordinates differ.

D.They are on the same axis because both

x

-coordinates are negative.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Equal $x$ -coordinates indicate that two points lie on the same vertical line. Here both points have $x=-7$ , while their $y$ -coordinates differ. Therefore, the locations are vertically aligned, not horizontally aligned.

Q13. A graph contains $A=(-2,5)$ , $B=(4,5)$ , and $C=(4,-1)$ . A route travels from $A$ to $B$ and then from $B$ to $C$ . Which statement correctly compares the two moves?

A.The first move is vertical and the second is horizontal.

B.The first move is 6 units horizontally and the second is 6 units vertically. ✅

C.Both moves are 6 units diagonally.

D.The first move is 4 units horizontally and the second is 5 units vertically.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: From $A$ to $B$ , the $y$ -coordinate remains $5$ while $x$ changes from $-2$ to $4$ , giving 6 horizontal units. From $B$ to $C$ , $x=4$ remains fixed while $y$ changes from $5$ to $-1$ , giving 6 vertical units.

Q14. A designer wants two points to be directly opposite across the origin, with neither point lying on an axis. If one point is $(3,-7)$ , which point satisfies the requirement?

(-3,7)

✅

(7,-3)

(-7,3)

(3,7)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Reflecting a point through the origin changes the sign of both coordinates while preserving their magnitudes. Thus $(3,-7)$ becomes $(-3,7)$ . The other choices either interchange coordinates or change only one sign.

Q15. Suppose four locations are $A=(-4,-2)$ , $B=(4,-2)$ , $C=(4,3)$ , and $D=(-4,3)$ . A student claims the figure cannot form a rectangle because some coordinates are negative. Which evaluation is correct?

A.The claim is correct because rectangles require positive coordinates.

B.The claim is correct because negative coordinates prevent equal side lengths.

C.The claim is incorrect because the points have two pairs of equal coordinates, creating horizontal and vertical sides. ✅

D.The claim is incorrect only if the origin lies inside the figure.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The negative coordinates do not prevent geometric shapes from forming. Here $A$ and $B$ share $y=-2$ , while $C$ and $D$ share $y=3$ . Also, $A,D$ share $x=-4$ and $B,C$ share $x=4$ , producing a rectangle.

Q16. A point $P=(x,y)$ is known to be in Quadrant IV and exactly 6 units from the horizontal axis. Which additional condition would uniquely determine $P$ ?

x>0

x=6

y=-6

✅

y>0

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Quadrant IV requires $x>0$ and $y<0$ . Being exactly 6 units from the horizontal axis means the vertical coordinate has magnitude 6. Combining these conditions gives $y=-6$ , while $x$ could still have many positive values, so the point is constrained to the line $y=-6$ , not uniquely to one coordinate unless an additional $x$ -condition is supplied. Among the choices, $y=-6$ is the necessary condition, whereas the others do not fully encode the stated information.

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