Lesson 2: Pythagorean Theorem Problem Solving - Maps and the Pythagorean Theorem

   Constructing Knowledge

Right triangles can be extracted from maps of city streets, country roads and other features. The right triangles can then be solved for unknown distances between points, using the Pythagorean Theorem. Conveniently, the base directions of north/south and east/west meet at a 90 degree angle.


Because many roads or sidewalks meet at 90° angles, a right angle triangle can be constructed by connecting opposite corners with a diagonal line.



Once a right angled triangle is drawn, the Pythagorean Theorem can be used to solve the distance of any of the sides.

EXAMPLE

Central Park in New York City, is one of the most famous parks in the world. The lengths of the streets that surround the park are 2.5 miles (4 km), for the streets that run north/south, and 0.5 miles (0.8 km), for the streets that run east/west.

Sally is located on the northwest corner of the park and needs to get to the southeast corner. How many fewer kilometres would she travel if she walked directly through the park, from the start to the finish instead of walking along the streets.

Solution


Step 1: Identify the given and the required values.

  • The north/south leg of the triangle is 4 km
  • The east/west leg of the triangle is 0.8 km
  • The distance through the park (hypotenuse) is the unknown.

Step 2: Draw and label the diagram.


Extract the portion of the picture that will enable the problem to be solved.



Step 3 and 4: Write the formula, substitute known values, and solve

\(\begin{align} a^2+b^2&=d^2 \\ \\ 4^2+0.8^2&=d^2 \\ \\ 16+0.64&=d^2 \\ \\ 16.64&=d^2 \\ \\ \sqrt{16.64}&=\sqrt{d^2} \\ \\ 4.079&=d \\ \end{align}\)

The distance through the park is approximately 4.1 km.

Step 5: Review the answer

The question asks how many fewer kilometres she would travel by walking through the park. To walk along the streets she would travel 4 km and 0.8 km.

street distance = 4 km + 0.8 km
= 4.8 km

The distance saved would be the distance along the streets less the distance through the park.

distance saved = 4.8 km - 4.1 km
= 0.7 km

Sally's walk would be 0.7 km shorter if she goes through the park.


Now, it is your turn! Complete the questions in your Chapter 3, Lesson 2 Practice Makes Perfect that refer to Maps and the Pythagorean Theorem.



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