Welcome, mathletes! This week's challenge is designed to test your problem-solving prowess and help you prepare for those tough MathCounts competitions. We'll tackle a problem that blends logic, geometry, and a bit of creative thinking – the perfect recipe for success!
This Week's MathCounts-Style Problem: The Tricky Triangle
A triangle has vertices at points A(1, 2), B(4, 6), and C(7, 2). Point D is located on side AB such that AD:DB = 1:2. Point E is located on side BC such that BE:EC = 1:2. What is the area of triangle ADE?
Think you've got the solution? Take your time; careful consideration is key to conquering this challenge. Before revealing the solution, let's explore some strategies that can help you approach similar problems in MathCounts competitions.
Strategies for Tackling Geometry Problems
Geometry problems often require a multi-step approach. Here's a breakdown of effective strategies:
1. Visualize and Sketch:
- Draw it out: Always begin by creating a clear diagram. Plot the points A, B, and C on a coordinate plane. This visual representation will help you understand the problem's geometry. Accurately plotting point D and E is crucial for success.
2. Utilize Coordinate Geometry:
- Section Formula: The given ratios (AD:DB = 1:2 and BE:EC = 1:2) indicate the use of the section formula. This formula allows you to find the coordinates of a point that divides a line segment in a given ratio. Remember the section formula: If a point P divides the line segment joining points (x₁, y₁) and (x₂, y₂) in the ratio m:n, then the coordinates of P are given by: x = (mx₂ + nx₁)/(m+n) y = (my₂ + ny₁)/(m+n)
3. Calculate Areas:
- Determinant Method: A powerful technique to find the area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is using the determinant method: Area = (1/2) |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
- Subtracting Areas: Once you have the coordinates of D and E, you can calculate the area of triangle ADE using the determinant method.
4. Check Your Work:
- Unit Consistency: Ensure that your units are consistent throughout your calculations.
- Reasonable Answer: Does your final answer seem logical considering the problem's context?
Solution to the Problem
Following the strategies outlined above:
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Plot the points: Create a graph showing points A, B, and C.
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Find D and E: Use the section formula to find the coordinates of D and E.
- For D (dividing AB in ratio 1:2): x = (14 + 21)/(1+2) = 2 y = (16 + 22)/(1+2) = 10/3
- For E (dividing BC in ratio 1:2): x = (17 + 24)/(1+2) = 15/3 = 5 y = (12 + 26)/(1+2) = 14/3
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Calculate Area of Triangle ADE: Use the determinant method with points A(1, 2), D(2, 10/3), and E(5, 14/3).
Area(ADE) = (1/2) |1(10/3 - 14/3) + 2(14/3 - 2) + 5(2 - 10/3)| = (1/2) |1(-4/3) + 2(8/3) + 5(-4/3)| = (1/2) |-4/3 + 16/3 - 20/3| = (1/2) |-8/3| = 4/3 square units.
Therefore, the area of triangle ADE is 4/3 square units.
Keep Practicing!
This problem demonstrates the importance of mastering coordinate geometry and area calculations. Remember to practice consistently, exploring different problem-solving approaches. The more you practice, the more confident and skilled you’ll become in tackling challenging MathCounts problems. Good luck!
