The Pythagorean Theorem

In mathematics, the Pythagorean theorem (or Pythagoras’s theorem) is a statement about the sides of a right triangle.

One of the angles of a right triangle is always equal to 90 degrees. This angle is called a right angle. The two sides next to the right angle are called the legs, and the other side is called the hypotenuse. The hypotenuse is the side opposite the right angle, and it is always the longest side.

The Pythagorean theorem states that the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. In Figure 1, the area of square “a” added to the area of square “b” equals the area of square ”c.”

If the lengths of the legs are “a” and “b” and the length of the hypotenuse is “c,” then:

[latex]\quad a^2 + b^2 = c^2[/latex]

In a practical application, if you know the lengths of two of the shorter sides (legs), you can solve for the hypotenuse by using the following equation:

[latex]\quad c = \sqrt{a^2 + b^2}[/latex]

If you know the lengths of one of the shorter sides and the hypotenuse, you can solve for the other shorter side by using the following equations:

[latex]\quad a = \sqrt{c^2 - b^2}\\ \quad\text{or}\\ \quad b = \sqrt{c^2 - a^2}[/latex]

Example 1:

Use the Pythagorean theorem to solve for the length of the pipe C shown in Figure 2, when A = 3′ and B = 7.25′.

Solution:

[latex]\quad \text{C} = \sqrt{3^2 + 7.25^2}\\ \quad \text{C} = \sqrt{9 + 52.56}\\ \quad \text{C} = \sqrt{61.56}\\ \quad \text{C} = 7.85\text{'}[/latex]

Example 2:

Using Figure 2 as your reference, solve for the length of the pipe B, when C = 55 cm and B = 47 cm.

Solution: 

[latex]\quad \text{A} = \sqrt{55^2 - 47^2}\\ \quad \text{A} = \sqrt{3,025 - 2,209}\\ \quad \text{A} = \sqrt{816}\\ \quad \text{A} = 28.57\text{ cm}[/latex]

Example 3:

Solve for the length of pipe B in Figure 2, when C = 18 m and A = 4.5 m.

Solution: 

[latex]\quad \text{B} = \sqrt{18^2 - 4.5^2}\\ \quad \text{B} = \sqrt{324 - 20.25}\\ \quad \text{B} = \sqrt{303.75}\\ \quad \text{B} = 17.43\text{ m}[/latex]

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Self-Test C-1.8: Use the Pythagorean Theorem of Right Angles

Complete Self-Test C-1.8 and check your answers.

Calculate the length of the missing side in the following right-angle triangles. Round your answers to two decimal places.

References

BCcampus. (n.d.). Playlist: Tools and equipment videos. BCcampus MediaSpace. https://media.bccampus.ca/playlist/details/0_3g8xp22x/categoryId/175673 Playlist Details – Trades Access Common Core Line C: Tools and Equipment Videos – BCcampus

BC Industry Training Authority. (2019). Piping trades apprenticeship program: Use Tools and Equipment—Level 1 harmonized [Binder]. Crown Publications, Queen’s Printer for British Columbia. https://www.crownpub.bc.ca/Product/Details/7960000261_S

• Plumber: Competency C-1 Use Mathematics and Science
• Steamfitter: Competency C-1 Use Mathematics and Science
• Sprinkler Fitter: Competency C-1 Use Mathematics and Science

Camosun College. (2019). Line C: Tools and Equipment—Competency D-1: Solve Trades Mathematical Problems (Rev. ed.) [Learning guide]. BCcampus.  https://collection.bccampus.ca/textbook/qFKGAJ78/

Camosun College. (2015). Trades Access Common Core Competency D-1: Solve Trades Mathematical Problems. Victoria, B.C.: Crown Publications. Download for free from the B.C. Open Textbook Collection (https://open.bccampus.ca/browse-ourcollection/find-open-textbooks/).

Camosun Innovates. (2022). Tools and Equipment Videos [Video playlist]. Camosun College/BCcampus. https://camosuninnovates.opened.ca/

Flinn, C. (n.d.). OER for Trades: Math for Trades [Video collection]. BCcampus MediaSpace. https://media.bccampus.ca/channel/OER%2Bfor%2BTrades%3A%2BMath%2Bfor%2BTrades/175670
Note: these videos align with the open textbooks Math for Trades: Volume 1 and Math for Trades: Volume 2. All videos are by Chad Flinn and available under a Creative Commons Attribution 4.0 Licence.:

Media Attributions

All figures are sourced from Industry Training Authority (2019) and/or Camosun College (2019) and are used under the Creative Commons Attribution 4.0 (CC BY 4.0) licence unless otherwise noted. Images copyrighted by the BC Industry Training Authority are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 (CC BY-NC-SA 4.0) licence.