Word Problems
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Geometry theorems appear frequently in real-world contexts and on standardized tests. This lesson helps you translate word problems into mathematical equations using triangle and circle theorems.
Word Problem Strategy
- Read carefully: Identify the geometric shapes and relationships described
- Draw a diagram: Sketch the situation and label known values
- Identify the theorem: Determine which theorem(s) apply
- Set up equations: Translate the words into mathematical expressions
- Solve and verify: Calculate and check that your answer makes sense
Common Word Problem Types
- Architecture: Roof angles, support structures, triangular designs
- Navigation: Direction changes, bearing angles
- Engineering: Circular components, inscribed shapes
- Art/Design: Symmetry, proportional triangles
Worked Examples
Example 1: Architecture
Problem: A triangular roof has two equal-length sides (isosceles). If the angle at the peak is 40 degrees, what angle does each side of the roof make with the horizontal base?
Solution:
- The roof is an isosceles triangle with vertex angle = 40 degrees
- The two base angles are equal (Isosceles Triangle Theorem)
- Sum of angles: 40 + x + x = 180
- 2x = 140, so x = 70 degrees
Answer: Each side makes a 70-degree angle with the base.
Example 2: Navigation
Problem: A ship changes course twice. If the first turn creates an exterior angle of 125 degrees with the original path, and the ship eventually returns to its original direction after the second turn, what angle must the second turn be?
Solution:
- The three directions form a triangle
- The exterior angle at the first turn is 125 degrees
- Interior angle at first turn: 180 - 125 = 55 degrees
- For the ship to return to original direction, the sum of the two turn angles equals the exterior angle
- Second turn: 125 - 55 = 70 degrees (or use: interior angles sum to 180)
Answer: The second turn must be 70 degrees.
Example 3: Circular Design
Problem: A circular garden has a decorative stone placed at the center (O) and two benches at points A and B on the edge. The angle AOB at the center is 100 degrees. A fountain at point C on the edge views both benches. What angle does the fountain make with the two benches (angle ACB)?
Solution:
- Central angle AOB = 100 degrees
- Inscribed angle ACB subtends the same arc AB
- By Inscribed Angle Theorem: ACB = (1/2) x AOB
- ACB = (1/2) x 100 = 50 degrees
Answer: The fountain views the benches at a 50-degree angle.
Practice Problems
Apply geometry theorems to solve these real-world scenarios.
Problem 1: A triangular park has angles of 65 degrees and 75 degrees at two corners. What is the measure of the third corner?
Solution: 65 + 75 + x = 180; x = 40 degrees
Answer: 40 degrees
Problem 2: An architect designs an isosceles triangular window. Each base angle must be 72 degrees to match the existing design. What must the vertex angle be?
Solution: 72 + 72 + vertex = 180; vertex = 36 degrees
Answer: 36 degrees
Problem 3: A surveyor measures an exterior angle of a triangular plot as 142 degrees. If one of the non-adjacent interior angles is 67 degrees, what is the other non-adjacent interior angle?
Solution: By Exterior Angle Theorem: 142 = 67 + x; x = 75 degrees
Answer: 75 degrees
Problem 4: A circular clock face has the center at O. At 2:00, the hour hand points to 2 and the minute hand points to 12. What inscribed angle would be formed by these positions as seen from any point on the clock face (not the center)?
Solution: The central angle is 60 degrees (2 hours x 30 degrees/hour).
Inscribed angle = (1/2) x 60 = 30 degrees
Answer: 30 degrees
Problem 5: A triangular sail has angles that are in the ratio 1:2:3. Find all three angles.
Solution: Let angles be x, 2x, and 3x. Then x + 2x + 3x = 180; 6x = 180; x = 30
Answer: 30 degrees, 60 degrees, and 90 degrees (a right triangle)
Problem 6: In a circular amphitheater, seats are arranged along the circumference. Two stage markers A and B are connected to the center O, forming a 96-degree angle. What angle does an audience member at position C on the circle see between the two markers?
Solution: By Inscribed Angle Theorem: angle ACB = (1/2) x 96 = 48 degrees
Answer: 48 degrees
Problem 7: A drone flies in a triangular path. It turns 130 degrees (exterior angle) at the first point and 145 degrees at the second point. What is the exterior angle at the third point?
Solution: Interior angles: 180-130=50 and 180-145=35. Third interior: 180-50-35=95. Third exterior: 180-95=85 degrees.
Answer: 85 degrees
Problem 8: An engineer inscribes a triangle in a circle such that one side passes through the center (diameter). What is the angle at the opposite vertex?
Solution: A diameter subtends a central angle of 180 degrees. By Thales' Theorem, the inscribed angle = 90 degrees.
Answer: 90 degrees
Problem 9: A tiled floor pattern uses congruent triangles. If the pattern requires that two angles of each triangle be 55 degrees and 63 degrees, what is the third angle?
Solution: 55 + 63 + x = 180; x = 62 degrees
Answer: 62 degrees
Problem 10: A pizza is cut so that the central angle of each slice is 45 degrees. If a customer sitting at the edge of a circular table looks at one slice, what angle does the slice's crust (arc) appear to subtend from their viewpoint?
Solution: The inscribed angle from any point on the circle = (1/2) x 45 = 22.5 degrees
Answer: 22.5 degrees
Check Your Understanding
- Why is drawing a diagram the first step in solving geometry word problems?
- How do you identify which theorem to use in a word problem?
- What real-world applications use the Inscribed Angle Theorem?
- How can you verify that your answer to a geometry word problem is reasonable?
Next Steps
- Practice translating more word problems into geometric diagrams
- Review any theorems that were difficult to apply
- Move on to Common Mistakes to learn what to avoid on tests