Explainer: Applications on the Pythagorean Theorem

In this explainer, we will learn how to apply the Pythagorean theorem in geometric questions and real-life situations.

The Pythagorean Theorem

The Pythagorean theorem states that, in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides.

If the hypotenuse is labeled β€œπ‘β€ and the two shorter sides are labeled β€œπ‘Žβ€ and β€œπ‘,” then the Pythagorean theorem states that π‘Ž2+𝑏2=𝑐2.

Once you know the theorem, the difficulty in solving more complex questions or problems is to identify that applying the Pythagorean theorem will allow us to answer the question and/or to extract from the question the useful information that allows drawing the right triangle.

Example 1: Using the Pythagorean Theorem to Solve Problems

The figure shows a student’s routes from home to school. Assuming that roads B and C are prependicular, what distance is saved if he takes road A instead of the other two?

Give your answer to two decimal places.

Answer

  1. Identify the question: how much shorter is road A compared to road B + road C?
  2. Identify what is given: road A (16.01 km) and road B (10 km) and road B is prependicular to road C.
  3. Identify a solving strategy:
    1. We need to find road C to answer the question. As roads B and C are perpendicular, the triangle formed by the three roads is a right triangle. It is then possible to apply the Pythagorean theorem to find road C.
    2. Then, to answer the question, we will calculate (road B + road C) – road A.
  4. Implement the strategy:
    • Part I:
      1. Roads B and C are the legs of the right triangle, and road A is the hypotenuse.
      2. With π‘Ž and 𝑏 the legs of the right triangle and 𝑐 the hypotenuse, the Pythagorean theorem states that π‘Ž2+𝑏2=𝑐2.
      3. This gives (roadB)2+(roadC)2=(roadA)2.
      Let π‘₯ be road C: 102+π‘₯2=16.012100+π‘₯2=256.3201π‘₯2=256.3210βˆ’100π‘₯2=156.3201.
      As π‘₯ is a length, it is positive, and so √156.3201β‰…12.50.
      Road A is approximately 12.50 km long.
    • Part II: (roadB+roadC)βˆ’roadAβ‰…(10+12.50)βˆ’16.01β‰…6.49. If the student takes the road A instead of road B and road C, he approximately saves 6.49 km.

Example 2: Using the Pythagorean Theorem to Solve Problems

A man on the top of a building wants to have a guy wire extended to a point on the ground 20 ft from the building. To the nearest foot, how long does the wire have to be if the building is 50 ft tall?

Answer

  1. Identify the question: find the length of the wire going from the ground to the top of the building.
  2. Identify what is given: the height of the building (50 ft) and the distance between where the wire is fastened to the ground and the building (20 ft).
  3. Identify a solving strategy: it is useful to draw a diagram at this stage.
    As the building and the ground are usually perpendicular, we see that the building’s vertical side, the ground, and the wire form a right triangle where the wire is the hypotenuse. We can apply the Pythagorean theorem in this triangle to find the length of the wire.
  4. Implement the strategy: In the right triangle identified above, the wire is the hypotenuse. With π‘Ž and 𝑏 the legs of the right triangle and 𝑐 the hypotenuse, the Pythagorean theorem states that π‘Ž2+𝑏2=𝑐2.

Here, the two legs of the triangle are 20 ft and 50 ft. Let π‘₯ be the length of the guy wire. By substituting these into the equation given by the Pythagorean theorem, we get 202+502=π‘₯2400+2,500=π‘₯22,900=π‘₯2.

As π‘₯ is a length, it is positive, and so π‘₯=√2,900β‰…53.85ft.

We are asked to give the answer to the nearest foot, so our answer is β€œthe wire has to be 54 feet long”.

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