Coordinate Distance - SHSAT Math

1. A rectangular playground has corners at $(2,1)$ and $(2,9)$ on the left side. How long is this side of the playground?
   Answer: 8 units When you see two points that form a side of a rectangle, you're finding the distance between those points. Since both corners have the same x-coordinate (2), this is a vertical line, making the calculation straightforward. To find the length of a vertical or horizontal line segment, you only need to look at the coordinate that changes. Here, the x-coordinate stays at 2, but the y-coordinates are 1 and 9. The distance is simply the difference between these y-values: $9 - 1 = 8$ units. You can verify this using the distance formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} = \sqrt{(2-2)^2 + (9-1)^2} = \sqrt{0 + 64} = 8$. Looking at the wrong answers: (A) 6 units might come from incorrectly subtracting coordinates, perhaps $9-2-1 = 6$. (B) 7 units could result from an off-by-one error, like counting grid squares instead of measuring distance. (D) 9 units is a common trap—this is just one of the y-coordinates, not the distance between them. Strategy tip: For distance problems involving points with the same x-coordinate or y-coordinate, take the shortcut—just subtract the coordinates that are different. Save the full distance formula for diagonal lines. Always double-check by visualizing: from y = 1 to y = 9 covers 8 units on the coordinate plane.
2. A submarine descends from $( -3, -120 )$ meters to $( -3, -350 )$ meters. How far did it travel?
   Answer: 230 m When you see coordinate points representing positions, you're dealing with distance calculations in a coordinate plane. The key insight here is recognizing what type of movement is happening and applying the appropriate distance formula. The submarine moves from $(-3, -120)$ to $(-3, -350)$. Notice that the x-coordinate stays the same at $-3$, meaning the submarine only moves vertically. When movement is purely horizontal or vertical, you don't need the full distance formula - just find the difference between the changing coordinates. Since only the y-coordinate changes, calculate: $|-350 - (-120)| = |-350 + 120| = |-230| = 230$ meters. The absolute value ensures you get a positive distance regardless of direction. Choice A (120 m) represents the absolute value of just the starting depth, ignoring the endpoint entirely. Choice C (350 m) is the absolute value of the final depth, again ignoring the starting position. Choice D (470 m) comes from incorrectly adding the absolute values of both coordinates: $|-120| + |-350| = 120 + 350 = 470$. This is a common trap - students sometimes add depths instead of finding the difference between positions. The correct answer is B (230 m). Remember: when finding distance between two points with the same x-coordinate or y-coordinate, you only need the difference of the changing coordinate. For diagonal movement, you'd need the full distance formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$, but vertical or horizontal movement simplifies to just the coordinate difference.
3. A crane moves a load from $( -5,8)$ straight down to $( -5, -12)$ and then straight right to $( 11, -12)$. What total vertical distance did the load travel?
   Answer: 20 units When you encounter coordinate geometry problems involving movement, focus on tracking each type of motion separately. This question asks specifically about vertical distance, so you need to identify which parts of the journey involve vertical movement. The crane moves from $(-5,8)$ to $(-5,-12)$, then to $(11,-12)$. Let's analyze each segment: First movement: $(-5,8)$ to $(-5,-12)$. Since the x-coordinate stays at $-5$, this is purely vertical movement. The load travels from a height of $8$ down to $-12$. The vertical distance is $|8 - (-12)| = |8 + 12| = 20$ units. Second movement: $(-5,-12)$ to $(11,-12)$. Since the y-coordinate stays at $-12$, this is purely horizontal movement. No vertical distance is traveled here. Total vertical distance = $20 + 0 = 20$ units. Choice A (12 units) incorrectly calculates just the distance from $0$ to $-12$, ignoring the initial height of $8$. Choice B (16 units) likely subtracts $8 - (-12) = 20$ but makes an arithmetic error, possibly confusing $8 + 12 = 20$ with $8 + 8 = 16$. Choice D (24 units) might incorrectly include some horizontal distance or add an extra calculation step. The key strategy here is to separate vertical from horizontal movement by examining which coordinate changes. When only the y-coordinate changes, you have vertical movement. Always use absolute value to find distance, since distance is always positive regardless of direction.
4. A warehouse coordinate grid shows Loading Dock at $( -8, 3)$. A truck parks directly south so the vertical distance is 18 units. What is the truck's coordinate?
   Answer: $(-8, -15)$ When you encounter coordinate geometry problems involving directional movement, focus on how each direction affects the x and y coordinates. Moving north or south only changes the y-coordinate, while moving east or west only changes the x-coordinate. The Loading Dock is at $(-8, 3)$. Since the truck parks "directly south," it moves straight down, keeping the same x-coordinate of $-8$. The vertical distance of 18 units means the truck moves 18 units down from the y-coordinate of 3. Moving south means subtracting from the y-coordinate: $3 - 18 = -15$. Therefore, the truck's position is $(-8, -15)$. Looking at the wrong answers: Choice B, $(10, 3)$, incorrectly changes the x-coordinate instead of the y-coordinate, suggesting movement east rather than south. Choice C, $(-8, 21)$, makes the common error of adding the distance instead of subtracting it—this would place the truck north of the dock, not south. Choice D, $(10, 21)$, combines both errors: changing the wrong coordinate and moving in the wrong direction. Remember this pattern: north/south movement affects only the y-coordinate (add for north, subtract for south), while east/west movement affects only the x-coordinate (add for east, subtract for west). Always double-check that your final coordinate reflects movement in the correct direction from the starting point.
5. Two shops on a street are represented by $(k,12)$ and $(k, -3)$. Their vertical separation is 15 units. Verify this statement.
   Answer: True, distance is 15 When you see two points with the same x-coordinate, you're looking at a vertical line situation. The distance between points that share an x-coordinate depends only on their y-coordinates. To find the vertical separation between $(k,12)$ and $(k,-3)$, you calculate the absolute difference of their y-coordinates: $|12 - (-3)| = |12 + 3| = 15$. Since both points have x-coordinate $k$, they lie on the same vertical line, making their separation purely vertical. The statement is correct. Choice A is right because the vertical distance is indeed 15 units, regardless of what value $k$ takes. Choice B suggests $k$ must be 0, but this reveals a misunderstanding. The x-coordinate $k$ can be any real number—it doesn't affect the vertical distance at all. Whether $k = 0$, $k = 100$, or $k = -50$, the vertical separation remains 15. Choice C claims the distance is $k + 15$, which incorrectly tries to incorporate the x-coordinate into a vertical distance calculation. This confuses horizontal position with vertical separation. Choice D states we need horizontal distance, but that's unnecessary. The question specifically asks about vertical separation, and when two points share the same x-coordinate, there is no horizontal distance between them. Remember: when points share the same x-coordinate, focus only on the y-coordinates to find vertical distance. The x-coordinate value is irrelevant to vertical separation calculations.
6. Use the figure shown. A robot starts at $(-10, 3)$ and moves according to this sequence: east 7 units, south 5 units, west 12 units, south 4 units, east 20 units. What is the horizontal distance between the robot's starting point and final position?
   Answer: $15$ Net east-west motion: $+7-12+20=+15$. Starting x-coordinate is $-10$, so final x-coordinate is $-10+15=5$. Horizontal distance $=|5-(-10)|=15$. B is just the final x-coordinate. C incorrectly adds only the eastward moves: $7+20=27$ (not 25). D is the net vertical motion: $-5-4=-9$.
7. Point $Z$ is at $(-2,7)$. Another point is 13 units directly north of $Z$. What is the $y$-coordinate of the other point?
   Answer: 20 This question tests coordinate geometry and directional movement on the coordinate plane. When you see "directly north," think about how compass directions translate to coordinate changes. Moving "directly north" means you're traveling straight up on the coordinate plane, which only affects the y-coordinate while keeping the x-coordinate unchanged. Starting from point Z at $(-2, 7)$, moving 13 units north means adding 13 to the y-coordinate: $7 + 13 = 20$. The other point is located at $(-2, 20)$, so its y-coordinate is 20. Let's examine why the other answers are incorrect. Choice A) -6 results from subtracting 13 instead of adding: $7 - 13 = -6$. This would be the y-coordinate if you moved 13 units south, not north. Choice B) 11 comes from a calculation error, possibly confusing the problem setup or misreading the starting y-coordinate. Choice D) 26 suggests adding 13 to both coordinates: $-2 + 13 = 11$ and $7 + 13 = 20$, then mistakenly using $11 + 15 = 26$ or some other computational error. Remember this key pattern: directional movements on coordinate planes follow specific rules. North/south movements only change y-coordinates (add for north, subtract for south), while east/west movements only change x-coordinates (add for east, subtract for west). Always identify which coordinate changes before performing any calculations.
8. Point A is at $(-3, 7)$ and point B is at $(5, 7)$. Point C is located on the same horizontal line as A and B, but is twice as far from A as B is from A. How many possible locations are there for point C?
   Answer: 2 Since A is at (-3, 7) and B is at (5, 7), the distance from A to B is |5 - (-3)| = 8 units. Point C must be on the same horizontal line (y = 7) and twice as far from A, so the distance from A to C is 16 units. C can be 16 units to the right of A at (-3 + 16, 7) = (13, 7), or 16 units to the left of A at (-3 - 16, 7) = (-19, 7). Therefore, there are 2 possible locations.
9. Use the coordinate plane shown. Point $A$ is at $(-9, 4)$. Point $B$ is obtained by reflecting $A$ across the vertical line $x = 2$. Point $C$ is obtained by reflecting $B$ across the horizontal line $y = -1$. What is the horizontal distance from $A$ to $C$?
   Answer: $22$ Reflecting $A(-9,4)$ across $x=2$: new x-coord is $2+(2-(-9))=13$, so $B(13,4)$. Reflecting $B$ across $y=-1$: x unchanged, so $C(13,-6)$. Horizontal distance from $A(-9,4)$ to $C(13,-6)$ is $|13-(-9)|=22$. B is half. C includes vertical. D comes from computing $2-(-9)-1=10$ error.
10. The points $L(a,5)$ and $M(a,-11)$ are 16 units apart. Which of the following describes all possible values of $a$?
    Answer: Any real number Both points have the same $x$-coordinate, $a$, so they lie on the same vertical line and the distance between them is just the difference in their $y$-coordinates. Distance $= |5 - (-11)| = 16$. This is 16 no matter what $a$ is: moving both points left or right together does not change how far apart they are. So the condition holds for every value of $a$, and $a$ can be any real number. "Cannot be determined" is tempting, but the set of possible values is known exactly: it is all real numbers. The value 0 is only one of those possible values, not all of them. The value 24 confuses a distance with a coordinate.
11. Point $M(5,\,-4)$ must be translated 11 units straight up. What will be the $y$-coordinate of the new point?
    Answer: 7 When you see a question about translating points on a coordinate plane, you're working with transformations that move points without changing their shape or size. Translation means sliding a point in a specific direction by a certain distance. To translate point $M(5, -4)$ straight up by 11 units, you need to add 11 to the $y$-coordinate while keeping the $x$-coordinate unchanged. Moving "up" on a coordinate plane means moving in the positive $y$-direction. Starting with $M(5, -4)$, the new $y$-coordinate becomes: $-4 + 11 = 7$. The $x$-coordinate stays at 5, so the new point is $(5, 7)$, making the $y$-coordinate 7. Looking at the wrong answers: Choice B (6) represents adding only 10 units instead of 11, likely from a calculation error. Choice C (5) confuses the $x$-coordinate with the $y$-coordinate—this is the original $x$-value, not what we're looking for. Choice D (-15) comes from subtracting 11 instead of adding it ($-4 - 11 = -15$), which would move the point down rather than up. The correct answer is A. Remember this pattern: when translating points, "up" or "down" changes only the $y$-coordinate (add for up, subtract for down), while "left" or "right" changes only the $x$-coordinate (add for right, subtract for left). Always double-check whether you're moving in the positive or negative direction.
12. Point $J(-1, y)$ is 8 units above $K(-1, -9)$. Find $y$.
    Answer: $-1$ When you see points with the same x-coordinate, you're dealing with a vertical line, which means you only need to focus on the y-coordinates to find distances. Since both points J(-1, y) and K(-1, -9) have the same x-coordinate of -1, they lie on a vertical line. The distance between them is simply the difference in their y-coordinates. When point J is "8 units above" point K, this means J's y-coordinate is 8 more than K's y-coordinate. Starting with K's y-coordinate of -9, we add 8 to find J's y-coordinate: $y = -9 + 8 = -1$. Therefore, $y = -1$. Let's check why the other answers don't work. Choice B) $-17$ would place point J at (-1, -17), which is actually 8 units below point K, not above it. Choice C) $-8$ would put J at (-1, -8), making it only 1 unit above K, not 8 units. Choice D) $7$ would place J at (-1, 7), which would be 16 units above K (since $7 - (-9) = 16$), not 8 units. The correct answer is A) $-1$. Strategy tip: When working with vertical distances on a coordinate plane, remember that "above" means adding to the y-coordinate, while "below" means subtracting. Also, don't let negative numbers confuse you—moving up 8 units from -9 still lands you at -1, not a positive number.
13. Refer to the map shown on the coordinate plane, where each unit represents $1$ mile. A park ranger walks from station $S(-4, 11)$ due south to a trailhead $T$, then due east to a lookout $L(9, 3)$. If the total distance walked is $21$ miles, what are the coordinates of trailhead $T$?
    Answer: $(-4, 3)$ Walking south means x stays $-4$. Walking east from $T$ to $L(9,3)$ means $T$ has y-coordinate $3$. So $T=(-4,3)$. Check: south distance $=|11-3|=8$; east distance $=|9-(-4)|=13$; total $=21$. ✓ B subtracts wrongly (11-13). C walks east first. D miscomputes.
14. A rectangle has vertices at $P(2, -1)$, $Q(2, 4)$, $R(7, 4)$, and $S(7, -1)$. If the rectangle is translated so that vertex P moves to $(-1, 3)$, what is the distance between the original position of vertex R and its new position?
    Answer: 5 units The translation moves P from (2, -1) to (-1, 3), which is a shift of (-3, 4). All vertices move by the same translation vector. R originally at (7, 4) moves to (7-3, 4+4) = (4, 8). The distance from original R(7, 4) to new R(4, 8) is calculated using vertical and horizontal distances: horizontal distance = |7-4| = 3, vertical distance = |8-4| = 4. Total distance = √(3² + 4²) = √(9 + 16) = √25 = 5 units.
15. Refer to the figure. On the coordinate plane, an 'L-shaped' region has vertices, in order: $(-4, 6)$, $(3, 6)$, $(3, 1)$, $(8, 1)$, $(8, -5)$, $(-4, -5)$. What is the total length of all horizontal edges of this region?
    Answer: $24$ Horizontal edges (consecutive vertices with the same y-coord): $(-4,6)\to(3,6)$: length $7$. $(3,1)\to(8,1)$: length $5$. $(8,-5)\to(-4,-5)$: length $12$. Total $=7+5+12=24$. A misses one. C omits middle edge. D adds vertical edges too.
16. If $S(0,\,12)$ is 9 units directly south of $T$, what are the coordinates of $T$?
    Answer: $(0,\,21)$ When working with coordinate geometry problems involving direction, you need to carefully track how movement affects the x and y coordinates. Moving north/south changes the y-coordinate, while moving east/west changes the x-coordinate. Since point $S(0, 12)$ is 9 units directly south of point $T$, this means $T$ is 9 units directly north of $S$. Moving north increases the y-coordinate while keeping the x-coordinate the same. Starting from $S(0, 12)$ and moving 9 units north: the x-coordinate stays 0, and the y-coordinate becomes $12 + 9 = 21$. Therefore, $T$ is at $(0, 21)$, which is choice B. Choice A, $(0, 3)$, represents moving 9 units south from $S$ instead of north. This is the result of subtracting 9 from the y-coordinate ($12 - 9 = 3$), which would place a point south of $S$, not north. Choice C, $(9, 12)$, incorrectly adds the distance to the x-coordinate instead of the y-coordinate. This represents moving 9 units east from $S$, not north. Choice D, $(-9, 12)$, also changes the x-coordinate instead of the y-coordinate, representing movement 9 units west from $S$. Remember: north/south movement only affects the y-coordinate (north = add, south = subtract), while east/west movement only affects the x-coordinate (east = add, west = subtract). Always double-check which direction the movement goes from the given point to the unknown point.
17. A vertical ladder on a grid leans against a wall from $(4,2)$ to $(4,18)$. What is the ladder's length on the grid?
    Answer: 16 units When you see coordinates and need to find distance, you're dealing with the distance formula or, in simpler cases like this one, basic coordinate geometry principles. The ladder extends from point $(4,2)$ to point $(4,18)$. Notice that both points have the same x-coordinate (4), meaning the ladder is perfectly vertical along the line $x = 4$. When two points share the same x-coordinate, finding the distance between them becomes much simpler than using the full distance formula. Since the ladder runs vertically from $(4,2)$ to $(4,18)$, you only need to find the difference in y-coordinates: $18 - 2 = 16$ units. This gives you the correct answer: B) 16 units. Let's examine why the other choices are incorrect. Choice A) 14 units might result from miscalculating $18 - 2$ or confusing this with another measurement. Choice C) 18 units represents a common trap—you might mistakenly think the ladder's length equals the higher y-coordinate, but this ignores that the ladder starts at $y = 2$, not the origin. Choice D) 20 units could come from incorrectly adding the coordinates somehow, like $18 + 2 = 20$. Remember: when finding distance between two points with the same x-coordinate, simply subtract the y-coordinates (larger minus smaller). When they share the same y-coordinate, subtract the x-coordinates. This shortcut saves time compared to the full distance formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ and reduces calculation errors.
18. A square has one side along the x-axis from $(1, 0)$ to $(6, 0)$. If another vertex of the square is at $(1, 5)$, what is the total distance traveled by a point that moves along the perimeter of the square from $(1, 0)$ to $(6, 5)$?
    Answer: 10 units The square has vertices at (1,0), (6,0), (6,5), and (1,5). Each side has length 5 units. To go from (1,0) to (6,5) along the perimeter, the path is: (1,0) → (1,5) → (6,5). The distance from (1,0) to (1,5) is 5 units vertically, and from (1,5) to (6,5) is 5 units horizontally. Total distance = 5 + 5 = 10 units.
19. Based on the coordinate plane shown, points $M(-7, k)$ and $N(5, k)$ are endpoints of a horizontal segment. Point $P(5, -3)$ lies directly below $N$, and $NP = 8$. If $k > 0$, what is the horizontal distance from $M$ to $N$?
    Answer: $12$ Since $NP = 8$ and $P$ is directly below $N$ at $y=-3$, $N$ has $y$-coordinate $-3+8=5$, so $k=5$. Horizontal distance from $M(-7,5)$ to $N(5,5)$ is $|5-(-7)|=12$. Distractor A is $|5-(-7)|$ computed incorrectly as $5-7\cdot (-1)$ error. C uses diagonal thinking. D is just $k$.
20. A point moves from $( -9, 4.5)$ to $( 7, 4.5)$ along a straight horizontal path. How many unit squares does it cross?
    Answer: 16 The path stays at $y = 4.5$, so it travels through the single row of unit squares between $y = 4$ and $y = 5$. The horizontal distance traveled is $7 - (-9) = 16$ units. Each unit of horizontal distance carries the point through exactly one unit square in that row, so it passes through 16 unit squares: 9 squares from $x = -9$ to $x = 0$ and 7 squares from $x = 0$ to $x = 7$. The answer 17 comes from counting the vertical grid lines from $x = -9$ to $x = 7$, inclusive, instead of the spaces between them, a classic fencepost error. The answers 13 and 14 undercount the squares on one side of the $y$-axis.

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