Mara's Run Calculating Total Distance With Pythagorean Theorem

#h1 Mara's Run A Mathematical Journey Home

In this article, we will delve into a classic problem involving distances and directions, perfect for sharpening your mathematical skills. We'll break down the problem step-by-step, offering a clear and engaging solution. This problem is a fantastic example of how the Pythagorean theorem can be applied in real-world scenarios. Let's embark on this mathematical journey together!

The Problem Unveiled

First, let's understand the problem, Mara's running journey. Mara embarked on a run that involved two distinct legs. Initially, she ran 3 kilometers in a northerly direction. Following this, she altered her course and ran 4 kilometers eastward. To complete her run, Mara chose the most direct path home, essentially forming the hypotenuse of a right-angled triangle. The core question we aim to answer is: What was the total distance Mara covered during her entire run?

This problem is a classic application of geometry and the Pythagorean theorem, which allows us to calculate distances in right-angled triangles. By visualizing Mara's run as a triangle, we can easily determine the length of the final leg of her journey and, consequently, the total distance she ran. This type of problem is not only a staple in mathematical education but also reflects real-world navigation and distance calculation scenarios.

Visualizing the Run

Before diving into calculations, visualizing the problem is key. Imagine Mara's journey as a right-angled triangle. The 3 km north run forms one leg of the triangle, and the 4 km east run forms the other leg. The direct path home is the hypotenuse, the longest side of the triangle, connecting her final position back to the starting point. This visualization is critical because it transforms the word problem into a geometric shape, making it easier to apply mathematical principles.

Drawing a simple diagram can greatly aid in this visualization. Sketch a north-south axis and an east-west axis. Plot Mara's 3 km northward run along the north axis and her 4 km eastward run along the east axis. Connect the starting and ending points to form the hypotenuse. Now, the problem is visually represented as a right-angled triangle, where the lengths of two sides are known, and we need to find the length of the third side. This step of visualization is not just helpful but often crucial in solving geometry-related problems.

Applying the Pythagorean Theorem

Now, let's put the Pythagorean theorem into action. This fundamental theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as: a² + b² = c², where 'a' and 'b' are the lengths of the two shorter sides (legs), and 'c' is the length of the hypotenuse.

In Mara's case, the 3 km northward run and the 4 km eastward run represent the two shorter sides of the triangle. So, we can substitute these values into the Pythagorean theorem: 3² + 4² = c². This equation allows us to directly calculate the length of the hypotenuse, which is the distance of Mara's direct path home. Understanding and applying the Pythagorean theorem is pivotal in solving this problem, highlighting its significance in geometry and practical distance calculations.

Calculating the Distance Home

With the Pythagorean theorem in place, let's calculate the distance Mara ran to return home. We have the equation: 3² + 4² = c². First, we square the known values: 3² equals 9, and 4² equals 16. Substituting these into the equation, we get 9 + 16 = c². Adding 9 and 16 gives us 25, so the equation simplifies to 25 = c².

To find the value of 'c', which represents the distance home, we need to take the square root of both sides of the equation. The square root of 25 is 5. Therefore, c = 5 km. This calculation reveals that Mara's direct path home was 5 kilometers long. This step demonstrates the practical application of the Pythagorean theorem in determining unknown distances, a skill valuable in various fields, including navigation and construction.

Finding the Total Distance

Now that we know the distance of Mara's direct path home, we can calculate the total distance of her run. Mara initially ran 3 kilometers north and then 4 kilometers east. We've just calculated that her direct path home was 5 kilometers. To find the total distance, we simply add these three distances together: 3 km + 4 km + 5 km.

Adding these values, we get a total distance of 12 kilometers. This is the total distance Mara covered during her entire run, including the initial legs and her direct return home. This final calculation answers the original question, providing a comprehensive solution to the problem. It also reinforces the importance of breaking down complex problems into smaller, manageable steps.

The Answer

Therefore, the total distance of Mara's run was 12 kilometers. This corresponds to option C in the multiple-choice answers provided. We arrived at this answer by first visualizing the problem as a right-angled triangle, then applying the Pythagorean theorem to find the distance of the direct path home, and finally, summing all the distances to calculate the total distance of the run.

This problem serves as an excellent illustration of how mathematical principles, like the Pythagorean theorem, can be used to solve practical problems involving distances and directions. Understanding these concepts is not only beneficial in academic settings but also in real-world scenarios, such as navigation, construction, and even everyday planning.

Why This Matters Real-World Applications

The problem we've solved isn't just a theoretical exercise. It has direct applications in various real-world scenarios. Understanding how to calculate distances using the Pythagorean theorem is crucial in fields like navigation, where determining the shortest path between two points is essential. Surveyors and construction workers use these principles to accurately measure distances and angles when laying out buildings or roads.

Furthermore, this type of problem enhances critical thinking and problem-solving skills. It teaches us how to break down a complex situation into simpler components, visualize geometric relationships, and apply mathematical formulas to find solutions. These skills are transferable to many areas of life, making the study of mathematics not just academically valuable but also practically useful.

Enhancing Problem-Solving Skills

This problem is a great example of how mathematical concepts can be applied to solve real-world situations. By visualizing Mara's run as a right triangle and applying the Pythagorean theorem, we were able to break down the problem into manageable steps. This approach to problem-solving is valuable in many areas of life.

To further enhance your problem-solving skills, consider practicing similar problems with varying distances and directions. Try changing the angles or adding more legs to the journey. The more you practice, the more comfortable you will become with applying mathematical concepts to solve complex problems. Remember, the key is to break down the problem, visualize the situation, and apply the appropriate formulas or theorems.

Conclusion

In conclusion, Mara's run was a total of 12 kilometers, calculated by understanding the geometry of the situation and applying the Pythagorean theorem. This exercise highlights the practical applications of mathematical concepts in everyday life. By breaking down the problem into smaller steps and visualizing the scenario, we were able to arrive at the correct solution. Remember, the ability to apply mathematical principles to real-world problems is a valuable skill that can be developed through practice and understanding.

This problem not only reinforces our understanding of the Pythagorean theorem but also emphasizes the importance of visualization and step-by-step problem-solving. As we've seen, mathematics is not just about formulas and equations; it's about understanding relationships and applying logical reasoning to find solutions. So, keep practicing, keep visualizing, and keep exploring the world of mathematics!