This article will explore various word problems that utilize the Pythagorean theorem, provide detailed solutions, and present an answer key for further understanding.
Understanding the Pythagorean Theorem
Before diving into the word problems, it is crucial to understand the components of the Pythagorean theorem:
- Right Triangle: A triangle with one angle measuring 90 degrees.
- Hypotenuse: The longest side of the right triangle, opposite the right angle.
- Legs: The other two sides of the triangle, which are perpendicular to each other.
Basic Formula
The basic formula used to solve problems involving the Pythagorean theorem is:
\[ c = \sqrt{a^2 + b^2} \]
Where:
- \(c\) = length of the hypotenuse
- \(a\) = length of one leg
- \(b\) = length of the other leg
Types of Word Problems
Word problems involving the Pythagorean theorem can generally be categorized into the following types:
1. Finding the length of a side: Given two sides of a triangle, find the length of the third side.
2. Distance problems: Involving distances between points in a coordinate plane or real-world scenarios.
3. Application problems: Real-life situations where the Pythagorean theorem is applicable.
Examples of Word Problems
Let’s explore a series of word problems that illustrate the application of the Pythagorean theorem.
Example Problem 1: Finding the Length of a Side
Problem: A right triangle has one leg that measures 6 cm and another leg that measures 8 cm. What is the length of the hypotenuse?
Solution:
Using the Pythagorean theorem:
1. Identify the lengths:
- \(a = 6\) cm
- \(b = 8\) cm
2. Apply the formula:
\[
c^2 = a^2 + b^2
\]
\[
c^2 = 6^2 + 8^2
\]
\[
c^2 = 36 + 64
\]
\[
c^2 = 100
\]
\[
c = \sqrt{100} = 10 \text{ cm}
\]
The length of the hypotenuse is 10 cm.
Example Problem 2: Distance Between Points
Problem: Find the distance between the points (3, 4) and (7, 1) in a coordinate plane.
Solution:
1. Calculate the differences in the x and y coordinates:
- \(a = 7 - 3 = 4\)
- \(b = 4 - 1 = 3\)
2. Apply the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
\[
c^2 = 4^2 + 3^2
\]
\[
c^2 = 16 + 9
\]
\[
c^2 = 25
\]
\[
c = \sqrt{25} = 5
\]
The distance between the points (3, 4) and (7, 1) is 5 units.
Example Problem 3: Application Problem
Problem: A ladder leans against a wall, forming a right triangle with the ground. If the base of the ladder is 4 feet away from the wall and the ladder reaches a height of 3 feet on the wall, what is the length of the ladder?
Solution:
1. Identify the legs of the triangle:
- One leg (\(a\)) = 3 feet (height)
- The other leg (\(b\)) = 4 feet (distance from the wall)
2. Apply the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
\[
c^2 = 3^2 + 4^2
\]
\[
c^2 = 9 + 16
\]
\[
c^2 = 25
\]
\[
c = \sqrt{25} = 5
\]
The length of the ladder is 5 feet.
Answer Key for Word Problems
Here is a concise answer key for the problems discussed:
1. Problem 1: Hypotenuse = 10 cm
2. Problem 2: Distance = 5 units
3. Problem 3: Length of the ladder = 5 feet
Practice Problems
To further practice using the Pythagorean theorem, consider the following problems:
1. A right triangle has legs of lengths 9 cm and 12 cm. Find the length of the hypotenuse.
2. Find the distance between the points (1, 2) and (4, 6).
3. A triangular park has a right angle where one side is 15 meters long and the other side is 20 meters long. What is the length of the diagonal path across the park?
Conclusion
Pythagorean theorem word problems are instrumental in understanding the application of geometry in real life. By breaking down the problems and applying the theorem step-by-step, students can develop a clearer comprehension of how to tackle various mathematical challenges. The answer key provided helps reinforce learning by allowing students to check their work and understand their mistakes. Mastery of the Pythagorean theorem not only aids in geometry but also lays a strong foundation for more advanced mathematical concepts.
Frequently Asked Questions
What is the Pythagorean theorem and how is it applied in word problems?
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (a² + b² = c²). In word problems, it's applied by identifying the lengths of the two legs of a right triangle and solving for the hypotenuse, or vice versa.
Can you provide an example of a word problem that uses the Pythagorean theorem?
Sure! A ladder is leaning against a wall. If the foot of the ladder is 4 feet away from the wall and the ladder is 10 feet long, how high does the ladder reach on the wall? Using the Pythagorean theorem, we can solve for the height (h): 4² + h² = 10², leading to h = 8 feet.
What are some common mistakes to avoid when solving Pythagorean theorem word problems?
Common mistakes include confusing the legs and the hypotenuse, miscalculating the squares of the lengths, or neglecting to check if the triangle is a right triangle before applying the theorem.
How do you determine which sides to use in a word problem involving the Pythagorean theorem?
Identify the right triangle within the problem and determine which two sides are the legs (perpendicular) and which side is the hypotenuse (the longest side opposite the right angle). Use the lengths of the legs to find the hypotenuse or vice versa.
What is the solution key for a word problem that asks for the distance between two points using the Pythagorean theorem?
To find the distance between two points (x1, y1) and (x2, y2), use the formula: distance = √((x2 - x1)² + (y2 - y1)²). This can be modeled as a right triangle where the difference in x-coordinates and y-coordinates represent the legs.
How can I practice Pythagorean theorem word problems effectively?
You can practice by solving a variety of real-world word problems, utilizing online resources, worksheets, and textbooks. Focus on problems that require you to visualize the triangle, identify known and unknown values, and apply the theorem correctly.
