IB Mathematics: Applications and Interpretation Quiz: Coordinate Geometry Applications
20 questions · exam conditions
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Coordinate Geometry ApplicationsQuestion 1 of 20

A circular irrigation system has its central pivot at C(5, -3) and has a radius of 8 units. A sensor is located at S(11, 4).

Determine the position of the sensor relative to the irrigated area.

The sensor is outside the irrigated area.
The sensor is inside the irrigated area.
The sensor is exactly on the edge of the irrigated area.
The position cannot be determined from the given information.
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IB Mathematics: Applications and Interpretation Quiz

IB Mathematics: Applications and Interpretation Quiz: Coordinate Geometry Applications

Practice Coordinate Geometry Applications in IB Mathematics: Applications and Interpretation with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Coordinate Geometry Applications, giving you a quick way to practice the rules, question types, and explanations that matter most for IB Mathematics: Applications and Interpretation.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A circular irrigation system has its central pivot at C(5, -3) and has a radius of 8 units. A sensor is located at S(11, 4).

Determine the position of the sensor relative to the irrigated area.

  1. The sensor is outside the irrigated area. (correct answer)
  2. The sensor is inside the irrigated area.
  3. The sensor is exactly on the edge of the irrigated area.
  4. The position cannot be determined from the given information.
Explanation: To determine the sensor's position, we must calculate the distance between the center of the system C(5, -3) and the sensor S(11, 4) and compare it to the radius of 8 units. Distance CS=(11−5)2+(4−(−3))2=62+72=36+49=85CS = \sqrt{(11-5)^2 + (4 - (-3))^2} = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85}. Since 85≈9.22\sqrt{85} \approx 9.22, and 9.22>89.22 > 8, the sensor is located outside the circular irrigated area.

Question 2

Two oil rigs are located at positions A(-6, -3) and B(10, 5) on a nautical chart. A supply boat travels from rig A to rig B.

The boat makes a scheduled stop at the midpoint of its journey to perform a maintenance check. What are the coordinates of this stopping point?

  1. (16, 8)
  2. (4, 2)
  3. (8, 4)
  4. (2, 1) (correct answer)
Explanation: When you encounter a problem asking for the midpoint between two locations, you're dealing with the midpoint formula, which finds the point exactly halfway between two given coordinates. The midpoint formula is: (x1+x22,y1+y22)\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) For oil rigs A(-6, -3) and B(10, 5), you calculate each coordinate of the midpoint separately: x-coordinate: −6+102=42=2\frac{-6 + 10}{2} = \frac{4}{2} = 2 y-coordinate: −3+52=22=1\frac{-3 + 5}{2} = \frac{2}{2} = 1 Therefore, the stopping point is at (2, 1), which is answer D. Looking at the incorrect options: Answer A (16, 8) appears to be the result of adding the coordinates instead of finding their average - this gives you (4, 2) when you add them, but (16, 8) might come from doubling that or making similar arithmetic errors. Answer B (4, 2) is exactly what you get if you add the x-coordinates and y-coordinates without dividing by 2 - a common mistake when students forget the "average" part of finding midpoints. Answer C (8, 4) is double the correct answer, suggesting someone found the midpoint correctly but then doubled it, perhaps confusing this with a different type of problem. Remember: the midpoint formula is essentially finding the average of the x-coordinates and the average of the y-coordinates. Think "average" whenever you see "midpoint" - this mental connection will help you avoid the trap of simply adding coordinates without dividing by 2.

Question 3

A city planning department uses a coordinate grid to map the locations of public services, where each unit represents one kilometre. A school is located at S(-4, 1) and a library is located at L(6, -5).

A new sports complex is planned to be built exactly halfway between the school and the library. What are the coordinates of the new sports complex?

  1. (-2, 1)
  2. (1, -2) (correct answer)
  3. (2, -4)
  4. (5, -3)
Explanation: The coordinates of the halfway point are found using the midpoint formula: M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right). Using the coordinates S(-4, 1) and L(6, -5): M=(−4+62,1+(−5)2)=(22,−42)=(1,−2)M = \left(\frac{-4 + 6}{2}, \frac{1 + (-5)}{2}\right) = \left(\frac{2}{2}, \frac{-4}{2}\right) = (1, -2).

Question 4

A warehouse is located at W(2, 2). A delivery truck travels to a store S. The distance between the warehouse and the store is exactly 58\sqrt{58} km. The store is located at S(x, 9).

Given that the store's x-coordinate is positive, find the value of x.

  1. 3
  2. 5 (correct answer)
  3. 7
  4. 9
Explanation: Using the distance formula, d2=(x2−x1)2+(y2−y1)2d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2. We have d=58d = \sqrt{58}, so d2=58d^2 = 58. The points are W(2, 2) and S(x, 9). Substitute these values into the formula: 58=(x−2)2+(9−2)258 = (x - 2)^2 + (9 - 2)^2. 58=(x−2)2+7258 = (x - 2)^2 + 7^2. 58=(x−2)2+4958 = (x - 2)^2 + 49. 9=(x−2)29 = (x - 2)^2. Taking the square root of both sides gives x−2=±3x - 2 = \pm 3. This leads to two solutions: x=2+3=5x = 2 + 3 = 5 or x=2−3=−1x = 2 - 3 = -1. The question specifies that the x-coordinate is positive, so we must choose x = 5.

Question 5

A triangular garden plot is defined by the vertices A(0, 0), B(6, 0), and C(3, 4).

What is the length of the median from vertex C to the side AB?

  1. 3.0
  2. 4.0 (correct answer)
  3. 5.0
  4. 6.0
Explanation: A median connects a vertex to the midpoint of the opposite side. First, find the midpoint of side AB. Let's call it M. M=(0+62,0+02)=(3,0)M = \left(\frac{0 + 6}{2}, \frac{0 + 0}{2}\right) = (3, 0). The median from vertex C is the line segment CM. Now, find the length of CM using the distance formula, with C(3, 4) and M(3, 0). d=(3−3)2+(4−0)2=02+42=16=4d = \sqrt{(3 - 3)^2 + (4 - 0)^2} = \sqrt{0^2 + 4^2} = \sqrt{16} = 4. The length of the median is 4.0 units.

Question 6

On a computer-aided design (CAD) drawing, a support beam is represented by the line segment PQ. One end of the beam is at P(3, 8). The center of the beam is at the midpoint M(-1, 5).

What are the coordinates of the other end of the beam, Q?

  1. (-5, 2) (correct answer)
  2. (1, 6.5)
  3. (2, 13)
  4. (7, 12)
Explanation: Let the coordinates of Q be (x,y)(x, y). The midpoint M is found by M=(xP+xQ2,yP+yQ2)M = \left(\frac{x_P + x_Q}{2}, \frac{y_P + y_Q}{2}\right). We can set up two equations: −1=3+x2-1 = \frac{3 + x}{2} and 5=8+y25 = \frac{8 + y}{2}. Solving the first equation: −2=3+x⇒x=−5-2 = 3 + x \Rightarrow x = -5. Solving the second equation: 10=8+y⇒y=210 = 8 + y \Rightarrow y = 2. Therefore, the coordinates of Q are (-5, 2).

Question 7

A ship's navigation system places it at coordinate A(2, 5). An iceberg is located at B(10, y). The ship's radar indicates the iceberg is exactly 10 nautical miles away. The grid units are in nautical miles.

Given that the iceberg's y-coordinate is positive and greater than the ship's y-coordinate, find the value of y.

  1. 6
  2. 8
  3. 11 (correct answer)
  4. 15
Explanation: Use the distance formula: d2=(x2−x1)2+(y2−y1)2d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2. We are given d=10d = 10, A(2, 5), and B(10, y). So, 102=(10−2)2+(y−5)210^2 = (10 - 2)^2 + (y - 5)^2. This simplifies to 100=82+(y−5)2100 = 8^2 + (y - 5)^2, which is 100=64+(y−5)2100 = 64 + (y - 5)^2. Subtracting 64 gives 36=(y−5)236 = (y - 5)^2. Taking the square root gives y−5=±6y - 5 = \pm 6. This leads to two possible solutions: y=5+6=11y = 5 + 6 = 11 or y=5−6=−1y = 5 - 6 = -1. The question states that y is positive and greater than 5, so we choose y = 11.

Question 8

A straight road connects Town A at (1, 10) and Town B at (13, 2). A service station, S, is located at the midpoint of the road. A new rest stop, R, is then built exactly halfway between Town A and the service station S.

What are the coordinates of the new rest stop R?

  1. (4, 8) (correct answer)
  2. (7, 6)
  3. (8.5, 5)
  4. (10, 4)
Explanation: This is a two-step problem. First, find the coordinates of the service station S, which is the midpoint of AB. S=(1+132,10+22)=(142,122)=(7,6)S = \left(\frac{1 + 13}{2}, \frac{10 + 2}{2}\right) = \left(\frac{14}{2}, \frac{12}{2}\right) = (7, 6). Next, find the coordinates of the rest stop R, which is the midpoint of AS. R=(1+72,10+62)=(82,162)=(4,8)R = \left(\frac{1 + 7}{2}, \frac{10 + 6}{2}\right) = \left(\frac{8}{2}, \frac{16}{2}\right) = (4, 8).

Question 9

A drone flies at a constant speed in a straight line. It starts at point P(12, 3) at time t=0 seconds and reaches point Q(-8, 23) at time t=20 seconds.

What are the coordinates of the drone at time t=10 seconds?

  1. (-2, 13)
  2. (2, 13) (correct answer)
  3. (4, 10)
  4. (22, -17)
Explanation: Since the drone flies at a constant speed, at half the total time (t=10 seconds is half of t=20 seconds), it will be at the midpoint of its journey between P and Q. We can find this location using the midpoint formula: M=(12+(−8)2,3+232)=(42,262)=(2,13)M = \left(\frac{12 + (-8)}{2}, \frac{3 + 23}{2}\right) = \left(\frac{4}{2}, \frac{26}{2}\right) = (2, 13).

Question 10

Three vertices of a parallelogram PQRS are P(-3, -2), Q(1, -5), and R(8, 1).

Assuming the vertices are listed in order, what are the coordinates of the fourth vertex, S?

  1. (12, -2)
  2. (5, -2)
  3. (6, 0)
  4. (4, 4) (correct answer)
Explanation: When you encounter a parallelogram problem with three given vertices, remember that opposite sides are parallel and equal in length. This means the vectors representing opposite sides are identical. In parallelogram PQRS, since the vertices are listed in order, we have sides PQ, QR, RS, and SP. The key insight is that vector PQ→\overrightarrow{PQ} equals vector SR→\overrightarrow{SR}, and vector QR→\overrightarrow{QR} equals vector PS→\overrightarrow{PS}. Let's use PQ→=SR→\overrightarrow{PQ} = \overrightarrow{SR}. First, calculate PQ→\overrightarrow{PQ}: PQ→=Q−P=(1,−5)−(−3,−2)=(4,−3)\overrightarrow{PQ} = Q - P = (1, -5) - (-3, -2) = (4, -3) Since SR→=PQ→\overrightarrow{SR} = \overrightarrow{PQ}, we have: SR→=R−S=(8,1)−S=(4,−3)\overrightarrow{SR} = R - S = (8, 1) - S = (4, -3) Solving for S: S=(8,1)−(4,−3)=(4,4)S = (8, 1) - (4, -3) = (4, 4) Let's examine why the other answers are incorrect. Answer A (12, -2) would result from incorrectly adding vectors instead of using the proper relationship. Answer B (5, -2) appears to come from a calculation error, possibly confusing the direction of one of the vectors. Answer C (6, 0) might result from averaging coordinates incorrectly or misunderstanding which sides are opposite in the parallelogram. Study tip: Always double-check your answer by verifying that both pairs of opposite sides are equal. Calculate QR→\overrightarrow{QR} and PS→\overrightarrow{PS} to confirm they're identical—this catches most errors and builds confidence in parallelogram problems.

Question 11

A warehouse is located at W(2, 2). A delivery truck travels to a store S. The distance between the warehouse and the store is exactly 58\sqrt{58} km. The store is located at S(x, 9).

Given that the store's x-coordinate is positive, find the value of x.

  1. 3
  2. 5 (correct answer)
  3. 7
  4. 9
Explanation: Using the distance formula, d2=(x2−x1)2+(y2−y1)2d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2. We have d=58d = \sqrt{58}, so d2=58d^2 = 58. The points are W(2, 2) and S(x, 9). Substitute these values into the formula: 58=(x−2)2+(9−2)258 = (x - 2)^2 + (9 - 2)^2. 58=(x−2)2+7258 = (x - 2)^2 + 7^2. 58=(x−2)2+4958 = (x - 2)^2 + 49. 9=(x−2)29 = (x - 2)^2. Taking the square root of both sides gives x−2=±3x - 2 = \pm 3. This leads to two solutions: x=2+3=5x = 2 + 3 = 5 or x=2−3=−1x = 2 - 3 = -1. The question specifies that the x-coordinate is positive, so we must choose x = 5.

Question 12

Three communication towers are located at points A(2, 2), B(8, 6), and C(5, 10) on a regional map grid.

To ensure signal overlap, the distances between the towers are calculated. What is the approximate perimeter of the triangle formed by these three towers?

  1. 19.6
  2. 20.8 (correct answer)
  3. 21.5
  4. 22.3
Explanation: Calculate each side using the distance formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. AB=(8−2)2+(6−2)2=36+16=52≈7.21AB = \sqrt{(8-2)^2 + (6-2)^2} = \sqrt{36+16} = \sqrt{52} \approx 7.21. BC=(5−8)2+(10−6)2=9+16=5BC = \sqrt{(5-8)^2 + (10-6)^2} = \sqrt{9+16} = 5. AC=(5−2)2+(10−2)2=9+64=73≈8.54AC = \sqrt{(5-2)^2 + (10-2)^2} = \sqrt{9+64} = \sqrt{73} \approx 8.54. Perimeter = 7.21+5+8.54=20.75≈20.87.21 + 5 + 8.54 = 20.75 \approx 20.8.

Question 13

A ship's navigation system places it at coordinate A(2, 5). An iceberg is located at B(10, y). The ship's radar indicates the iceberg is exactly 10 nautical miles away. The grid units are in nautical miles.

Given that the iceberg's y-coordinate is positive and greater than the ship's y-coordinate, find the value of y.

  1. 6
  2. 8
  3. 11 (correct answer)
  4. 15
Explanation: Use the distance formula: d2=(x2−x1)2+(y2−y1)2d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2. We are given d=10d = 10, A(2, 5), and B(10, y). So, 102=(10−2)2+(y−5)210^2 = (10 - 2)^2 + (y - 5)^2. This simplifies to 100=82+(y−5)2100 = 8^2 + (y - 5)^2, which is 100=64+(y−5)2100 = 64 + (y - 5)^2. Subtracting 64 gives 36=(y−5)236 = (y - 5)^2. Taking the square root gives y−5=±6y - 5 = \pm 6. This leads to two possible solutions: y=5+6=11y = 5 + 6 = 11 or y=5−6=−1y = 5 - 6 = -1. The question states that y is positive and greater than 5, so we choose y = 11.

Question 14

A landscape designer is mapping out a rectangular garden on a grid. The corners of the garden are at A(1, 2), B(7, 2), C(7, 10), and D(1, 10). A fountain is to be placed at the intersection of the diagonals of the garden.

What are the coordinates of the location for the fountain?

  1. (3, 5)
  2. (4, 6) (correct answer)
  3. (5, 7)
  4. (8, 12)
Explanation: The diagonals of a rectangle bisect each other, so their intersection point is the midpoint of either diagonal. We can use the coordinates of the opposite vertices A(1, 2) and C(7, 10) to find the midpoint. Using the midpoint formula: M=(1+72,2+102)=(82,122)=(4,6)M = \left(\frac{1 + 7}{2}, \frac{2 + 10}{2}\right) = \left(\frac{8}{2}, \frac{12}{2}\right) = (4, 6). The same result would be found using the midpoint of B and D.

Question 15

Two hikers start at the same base camp. Hiker 1 walks to a viewpoint at V(5, 8). Hiker 2 walks to a lake at L(-3, 2). The coordinates are given in kilometres.

How much further did Hiker 1 walk than Hiker 2, assuming they walked in straight lines from a base camp at (0,0)?

  1. 5.82 km (correct answer)
  2. 6.10 km
  3. 9.43 km
  4. 13.04 km
Explanation: First, calculate the distance for Hiker 1 from (0,0) to V(5,8): d1=(5−0)2+(8−0)2=25+64=89≈9.434d_1 = \sqrt{(5-0)^2 + (8-0)^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.434 km. Next, calculate the distance for Hiker 2 from (0,0) to L(-3,2): d2=(−3−0)2+(2−0)2=9+4=13≈3.606d_2 = \sqrt{(-3-0)^2 + (2-0)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.606 km. The difference in distance is d1−d2=9.434−3.606=5.828d_1 - d_2 = 9.434 - 3.606 = 5.828 km. The closest answer is 5.82 km.

Question 16

A straight road connects Town A at (1, 10) and Town B at (13, 2). A service station, S, is located at the midpoint of the road. A new rest stop, R, is then built exactly halfway between Town A and the service station S.

What are the coordinates of the new rest stop R?

  1. (4, 8) (correct answer)
  2. (7, 6)
  3. (8.5, 5)
  4. (10, 4)
Explanation: This is a two-step problem. First, find the coordinates of the service station S, which is the midpoint of AB. S=(1+132,10+22)=(142,122)=(7,6)S = \left(\frac{1 + 13}{2}, \frac{10 + 2}{2}\right) = \left(\frac{14}{2}, \frac{12}{2}\right) = (7, 6). Next, find the coordinates of the rest stop R, which is the midpoint of AS. R=(1+72,10+62)=(82,162)=(4,8)R = \left(\frac{1 + 7}{2}, \frac{10 + 6}{2}\right) = \left(\frac{8}{2}, \frac{16}{2}\right) = (4, 8).

Question 17

A drone flies at a constant speed in a straight line. It starts at point P(12, 3) at time t=0 seconds and reaches point Q(-8, 23) at time t=20 seconds.

What are the coordinates of the drone at time t=10 seconds?

  1. (-2, 13)
  2. (2, 13) (correct answer)
  3. (4, 10)
  4. (22, -17)
Explanation: Since the drone flies at a constant speed, at half the total time (t=10 seconds is half of t=20 seconds), it will be at the midpoint of its journey between P and Q. We can find this location using the midpoint formula: M=(12+(−8)2,3+232)=(42,262)=(2,13)M = \left(\frac{12 + (-8)}{2}, \frac{3 + 23}{2}\right) = \left(\frac{4}{2}, \frac{26}{2}\right) = (2, 13).

Question 18

A map is drawn on a coordinate grid where 1 unit represents 50 metres. A straight fence is to be built between point A(-7, 2) and point B(5, 7). The cost of the fence is $40 per metre.

What is the total cost to build the fence?

  1. $2,600
  2. $13,000
  3. $26,000 (correct answer)
  4. $32,500
Explanation: First, find the distance between A and B in grid units: d=(5−(−7))2+(7−2)2=122+52=144+25=169=13d = \sqrt{(5 - (-7))^2 + (7 - 2)^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 units. Next, convert this distance to metres using the scale: Length = 13 units×50 m/unit=65013 \text{ units} \times 50 \text{ m/unit} = 650 metres. Finally, calculate the total cost: Cost = 650 \text{ m} \times \40/\text{m} = $26,000$.

Question 19

Three vertices of a parallelogram PQRS are P(-3, -2), Q(1, -5), and R(8, 1).

Assuming the vertices are listed in order, what are the coordinates of the fourth vertex, S?

  1. (12, -2)
  2. (5, -2)
  3. (6, 0)
  4. (4, 4) (correct answer)
Explanation: When you encounter a parallelogram problem with three given vertices, remember that opposite sides are parallel and equal in length. This means the vectors representing opposite sides are identical. In parallelogram PQRS, since the vertices are listed in order, we have sides PQ, QR, RS, and SP. The key insight is that vector PQ→\overrightarrow{PQ} equals vector SR→\overrightarrow{SR}, and vector QR→\overrightarrow{QR} equals vector PS→\overrightarrow{PS}. Let's use PQ→=SR→\overrightarrow{PQ} = \overrightarrow{SR}. First, calculate PQ→\overrightarrow{PQ}: PQ→=Q−P=(1,−5)−(−3,−2)=(4,−3)\overrightarrow{PQ} = Q - P = (1, -5) - (-3, -2) = (4, -3) Since SR→=PQ→\overrightarrow{SR} = \overrightarrow{PQ}, we have: SR→=R−S=(8,1)−S=(4,−3)\overrightarrow{SR} = R - S = (8, 1) - S = (4, -3) Solving for S: S=(8,1)−(4,−3)=(4,4)S = (8, 1) - (4, -3) = (4, 4) Let's examine why the other answers are incorrect. Answer A (12, -2) would result from incorrectly adding vectors instead of using the proper relationship. Answer B (5, -2) appears to come from a calculation error, possibly confusing the direction of one of the vectors. Answer C (6, 0) might result from averaging coordinates incorrectly or misunderstanding which sides are opposite in the parallelogram. Study tip: Always double-check your answer by verifying that both pairs of opposite sides are equal. Calculate QR→\overrightarrow{QR} and PS→\overrightarrow{PS} to confirm they're identical—this catches most errors and builds confidence in parallelogram problems.

Question 20

A robot is programmed to move from an initial position A to a final position B. The midpoint of this path is M(4, -1). The final position is B(9, -7).

What was the robot's initial position A?

  1. (14, -13)
  2. (1, -5)
  3. (6.5, -4)
  4. (-1, 5) (correct answer)
Explanation: When you encounter midpoint problems, you're working with the coordinate relationship between three points. The midpoint formula connects an initial point, final point, and the point exactly halfway between them. Given that M(4, -1) is the midpoint between A and B(9, -7), you can use the midpoint formula: M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right). Let A have coordinates (xA,yA)(x_A, y_A). Then: 4=xA+924 = \frac{x_A + 9}{2} and −1=yA+(−7)2-1 = \frac{y_A + (-7)}{2} Solving the first equation: 8=xA+98 = x_A + 9, so xA=−1x_A = -1 Solving the second equation: −2=yA−7-2 = y_A - 7, so yA=5y_A = 5 Therefore, A(-1, 5), which is answer D. Let's check why the other options fail. Option A (14, -13) gives a midpoint of (14+92,−13+(−7)2)=(11.5,−10)\left(\frac{14+9}{2}, \frac{-13+(-7)}{2}\right) = (11.5, -10), not (4, -1). Option B (1, -5) yields (1+92,−5+(−7)2)=(5,−6)\left(\frac{1+9}{2}, \frac{-5+(-7)}{2}\right) = (5, -6). Option C (6.5, -4) produces (6.5+92,−4+(−7)2)=(7.75,−5.5)\left(\frac{6.5+9}{2}, \frac{-4+(-7)}{2}\right) = (7.75, -5.5). The key strategy here is to remember that if you know the midpoint and one endpoint, you can find the other endpoint by rearranging the midpoint formula. Many students make arithmetic errors when working backwards from the midpoint, so always verify your answer by checking that it produces the given midpoint when paired with the known endpoint.