Grade: Grade 10 Subject: Mathematics Unit: Coordinate Geometry Lesson: 4 of 6 SAT: Geometry+Trigonometry ACT: Math

Word Problems

📖 Learn

This lesson focuses on applying coordinate geometry to real-world situations. Word problems require you to translate scenarios into mathematical models using the coordinate plane.

Strategies for Word Problems

  1. Read carefully - Identify what is given and what is asked
  2. Draw a diagram - Sketch the situation on a coordinate plane
  3. Assign coordinates - Place points strategically (often using the origin)
  4. Choose the right formula - Distance, midpoint, or slope
  5. Solve and interpret - Make sure your answer makes sense in context

Common Word Problem Types

  • Navigation/Maps: Finding distances between locations
  • Construction: Finding midpoints for supports or center points
  • Motion: Determining if paths are parallel or perpendicular
  • Area/Perimeter: Calculating measurements of figures on a coordinate plane

💡 Examples

Work through these examples to see how to approach word problems.

Example 1: Map Distance

Problem: On a city map where each unit represents 1 mile, a hospital is located at (3, 7) and a fire station is at (15, 2). How far apart are they?

Solution:

  1. This is a distance problem between two points
  2. d = sqrt[(15-3)^2 + (2-7)^2]
  3. d = sqrt[144 + 25] = sqrt[169] = 13
  4. Answer: The hospital and fire station are 13 miles apart

Example 2: Construction Project

Problem: A suspension bridge has support towers at coordinates A(20, 0) and B(80, 0). Where should a center support be placed?

Solution:

  1. This is a midpoint problem
  2. M = ((20+80)/2, (0+0)/2)
  3. M = (50, 0)
  4. Answer: The center support should be at (50, 0), which is 50 units from the origin along the x-axis

Example 3: Garden Design

Problem: A rectangular garden has corners at A(0, 0), B(8, 0), C(8, 6), and D(0, 6). Find the length of the diagonal and verify it is a rectangle.

Solution:

  1. Diagonal AC: d = sqrt[(8-0)^2 + (6-0)^2] = sqrt[64 + 36] = sqrt[100] = 10
  2. To verify rectangle: opposite sides must be equal and parallel
    • AB = 8, DC = 8 (both horizontal, parallel)
    • AD = 6, BC = 6 (both vertical, parallel)
  3. Answer: Diagonal is 10 units; it is a valid rectangle

✏️ Practice

Solve these 10 word problems. Draw diagrams to help visualize each situation.

1. A drone starts at position (0, 0) and flies to (9, 12). If each unit is 100 meters, how far did the drone travel?

2. Two friends are at a park. Alex is at coordinates (2, 5) and Jordan is at (10, 11). They want to meet exactly halfway. Where should they meet?

3. A ship travels from port A at (0, 0) to port B at (40, 30). If the ship can travel 10 km per hour, how long will the trip take?

4. A triangular plot of land has vertices at P(0, 0), Q(12, 0), and R(6, 8). What is the perimeter of the plot?

5. A sprinkler at the origin can spray water in a circle with radius 15 feet. Can it reach a flower bed at position (9, 12)?

6. A cable company needs to run a cable from point A(-3, 4) to point B(5, -2). If cable costs $2.50 per unit length, how much will the cable cost?

7. A treasure map shows buried treasure at the midpoint between a palm tree at (8, 2) and a rock at (4, 10). Where is the treasure?

8. A road runs through points (0, 3) and (6, 7). A second road runs through (2, 0) and (8, 4). Are these roads parallel? Explain.

9. A circular track has its center at (5, 5). If a point on the track is at (5, 12), what is the circumference of the track?

10. A kite has vertices at A(0, 3), B(2, 5), C(4, 3), and D(2, -1). Find the lengths of both diagonals AC and BD.

✅ Check Your Understanding

Verify your answers to the practice problems.

1. 1500 meters (15 units x 100 m)

2. (6, 8)

3. 5 hours (50 km at 10 km/hr)

4. 32 units (12 + 10 + 10)

5. Yes (distance is 15 feet, equal to radius)

6. $25.00 (10 units x $2.50)

7. (6, 6)

8. Yes, both have slope 2/3

9. 14pi (approximately 43.98) units; radius = 7

10. AC = 4 units, BD = 6 units

🚀 Next Steps

  • Practice drawing coordinate diagrams for each problem type
  • Review problems you found challenging
  • Look for real-world applications of coordinate geometry in your daily life
  • Move on to Common Mistakes to learn what errors to avoid