Coordinate Geometry Applications - IB Mathematics Applications and Interpretation

1. What is the length of the median from vertex C to the side AB?
   Answer: 4.0 A median connects a vertex to the midpoint of the opposite side. First, find the midpoint of side AB. Let's call it M. $M = \left(\frac{0 + 6}{2}, \frac{0 + 0}{2}\right) = (3, 0)$. 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). $d = \sqrt{(3 - 3)^2 + (4 - 0)^2} = \sqrt{0^2 + 4^2} = \sqrt{16} = 4$. The length of the median is 4.0 units.
2. What are the coordinates of the location for the fountain?
   Answer: (4, 6) 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: $M = \left(\frac{1 + 7}{2}, \frac{2 + 10}{2}\right) = \left(\frac{8}{2}, \frac{12}{2}\right) = (4, 6)$. The same result would be found using the midpoint of B and D.
3. Assuming the vertices are listed in order, what are the coordinates of the fourth vertex, S?
   Answer: (4, 4) 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 $\overrightarrow{PQ}$ equals vector $\overrightarrow{SR}$, and vector $\overrightarrow{QR}$ equals vector $\overrightarrow{PS}$. Let's use $\overrightarrow{PQ} = \overrightarrow{SR}$. First, calculate $\overrightarrow{PQ}$: $\overrightarrow{PQ} = Q - P = (1, -5) - (-3, -2) = (4, -3)$ Since $\overrightarrow{SR} = \overrightarrow{PQ}$, we have: $\overrightarrow{SR} = R - S = (8, 1) - S = (4, -3)$ Solving for S: $S = (8, 1) - (4, -3) = (4, 4)$ 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 $\overrightarrow{QR}$ and $\overrightarrow{PS}$ to confirm they're identical—this catches most errors and builds confidence in parallelogram problems.
4. Given that the iceberg's y-coordinate is positive and greater than the ship's y-coordinate, find the value of y.
   Answer: 11 Use the distance formula: $d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$. We are given $d = 10$, A(2, 5), and B(10, y). So, $10^2 = (10 - 2)^2 + (y - 5)^2$. This simplifies to $100 = 8^2 + (y - 5)^2$, which is $100 = 64 + (y - 5)^2$. Subtracting 64 gives $36 = (y - 5)^2$. Taking the square root gives $y - 5 = \pm 6$. This leads to two possible solutions: $y = 5 + 6 = 11$ or $y = 5 - 6 = -1$. The question states that y is positive and greater than 5, so we choose y = 11.
5. What are the coordinates of the drone at time t=10 seconds?
   Answer: (2, 13) 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: $M = \left(\frac{12 + (-8)}{2}, \frac{3 + 23}{2}\right) = \left(\frac{4}{2}, \frac{26}{2}\right) = (2, 13)$.
6. What was the robot's initial position A?
   Answer: (-1, 5) 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: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. Let A have coordinates $(x_A, y_A)$. Then: $4 = \frac{x_A + 9}{2}$ and $-1 = \frac{y_A + (-7)}{2}$ Solving the first equation: $8 = x_A + 9$, so $x_A = -1$ Solving the second equation: $-2 = y_A - 7$, so $y_A = 5$ Therefore, A(-1, 5), which is answer D. Let's check why the other options fail. Option A (14, -13) gives a midpoint of $\left(\frac{14+9}{2}, \frac{-13+(-7)}{2}\right) = (11.5, -10)$, not (4, -1). Option B (1, -5) yields $\left(\frac{1+9}{2}, \frac{-5+(-7)}{2}\right) = (5, -6)$. Option C (6.5, -4) produces $\left(\frac{6.5+9}{2}, \frac{-4+(-7)}{2}\right) = (7.75, -5.5)$. 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.
7. What are the coordinates of the other end of the beam, Q?
   Answer: (-5, 2) Let the coordinates of Q be $(x, y)$. The midpoint M is found by $M = \left(\frac{x_P + x_Q}{2}, \frac{y_P + y_Q}{2}\right)$. We can set up two equations: $-1 = \frac{3 + x}{2}$ and $5 = \frac{8 + y}{2}$. Solving the first equation: $-2 = 3 + x \Rightarrow x = -5$. Solving the second equation: $10 = 8 + y \Rightarrow y = 2$. Therefore, the coordinates of Q are (-5, 2).
8. What are the coordinates of the new rest stop R?
   Answer: (4, 8) This is a two-step problem. First, find the coordinates of the service station S, which is the midpoint of AB. $S = \left(\frac{1 + 13}{2}, \frac{10 + 2}{2}\right) = \left(\frac{14}{2}, \frac{12}{2}\right) = (7, 6)$. Next, find the coordinates of the rest stop R, which is the midpoint of AS. $R = \left(\frac{1 + 7}{2}, \frac{10 + 6}{2}\right) = \left(\frac{8}{2}, \frac{16}{2}\right) = (4, 8)$.
9. 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?
   Answer: (1, -2) The coordinates of the halfway point are found using the midpoint formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. Using the coordinates S(-4, 1) and L(6, -5): $M = \left(\frac{-4 + 6}{2}, \frac{1 + (-5)}{2}\right) = \left(\frac{2}{2}, \frac{-4}{2}\right) = (1, -2)$.
10. 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?
    Answer: (2, 1) 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: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ For oil rigs A(-6, -3) and B(10, 5), you calculate each coordinate of the midpoint separately: x-coordinate: $\frac{-6 + 10}{2} = \frac{4}{2} = 2$ y-coordinate: $\frac{-3 + 5}{2} = \frac{2}{2} = 1$ 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.
11. What is the total cost to build the fence?
    Answer: $26,000 First, find the distance between A and B in grid units: $d = \sqrt{(5 - (-7))^2 + (7 - 2)^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13$ units. Next, convert this distance to metres using the scale: Length = $13 \text{ units} \times 50 \text{ m/unit} = 650$ metres. Finally, calculate the total cost: Cost = $650 \text{ m} \times \$40/\text{m} = \$26,000$.
12. Determine the position of the sensor relative to the irrigated area.
    Answer: The sensor is outside the irrigated area. 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 $CS = \sqrt{(11-5)^2 + (4 - (-3))^2} = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85}$. Since $\sqrt{85} \approx 9.22$, and $9.22 > 8$, the sensor is located outside the circular irrigated area.
13. Given that the store's x-coordinate is positive, find the value of x.
    Answer: 5 Using the distance formula, $d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$. We have $d = \sqrt{58}$, so $d^2 = 58$. The points are W(2, 2) and S(x, 9). Substitute these values into the formula: $58 = (x - 2)^2 + (9 - 2)^2$. $58 = (x - 2)^2 + 7^2$. $58 = (x - 2)^2 + 49$. $9 = (x - 2)^2$. Taking the square root of both sides gives $x - 2 = \pm 3$. This leads to two solutions: $x = 2 + 3 = 5$ or $x = 2 - 3 = -1$. The question specifies that the x-coordinate is positive, so we must choose x = 5.
14. How much further did Hiker 1 walk than Hiker 2, assuming they walked in straight lines from a base camp at (0,0)?
    Answer: 5.82 km First, calculate the distance for Hiker 1 from (0,0) to V(5,8): $d_1 = \sqrt{(5-0)^2 + (8-0)^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.434$ km. Next, calculate the distance for Hiker 2 from (0,0) to L(-3,2): $d_2 = \sqrt{(-3-0)^2 + (2-0)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.606$ km. The difference in distance is $d_1 - d_2 = 9.434 - 3.606 = 5.828$ km. The closest answer is 5.82 km.
15. To ensure signal overlap, the distances between the towers are calculated. What is the approximate perimeter of the triangle formed by these three towers?
    Answer: 20.8 Calculate each side using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. $AB = \sqrt{(8-2)^2 + (6-2)^2} = \sqrt{36+16} = \sqrt{52} \approx 7.21$. $BC = \sqrt{(5-8)^2 + (10-6)^2} = \sqrt{9+16} = 5$. $AC = \sqrt{(5-2)^2 + (10-2)^2} = \sqrt{9+64} = \sqrt{73} \approx 8.54$. Perimeter = $7.21 + 5 + 8.54 = 20.75 \approx 20.8$.

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