E2.4 Describe the Pythagorean relationship using various geometric models, and apply the theorem to solve problems involving an unknown side length for a given right triangle.
Activity 1: Pythagoras: A Puzzle!
Image A puzzle made of various polygons. The puzzle consists of 2 large squares, that are made of irregular trapezoids. The 2 squares are joined by a triangle and at the end of the triangle there is a square.
Provide students with the puzzle below or create a puzzle using dynamic geometry software. Upload a document to a web platform so that students can record their observations.
Ask the students to put the pieces of the puzzle together, taking into account the following instructions:
- Choose the triangle, then place it in the middle of your desk.
- Assemble some of the puzzle pieces around the triangle to form a square on each side of the shape.
- What are your observations?
- Share your observations with another student or record them in the uploaded document.
- What are the characteristics of the triangle?
- If the measurements of the triangle were 3 cm, 4 cm, and 5 cm, what relationship could you establish based on your observations?
- If the measurements of the triangle were 5 cm, 8 cm, and 10 cm, would you be able to make a relationship based on your observations?
- What conclusions can you draw from this activity?
Skills to be developed related to spatial visualization:
- Constancy of shapes
- Perception of positions
- Visual discrimination
- Mental rotation
Source: translated from Guide d’enseignement efficace des mathématiques, de la 7e à la 10e année, Mesure et géométrie, Fascicule 3, p. 14
Activity 2: Problem Related to the Pythagorean Theorem
- When preparing for the county fair, Amarok wants to install a wire at the top of the main pole of a large tent so that it is securely anchored in the ground. If the pole is 16 metres long and she wants to anchor the wire 5 metres from its base, what should be the minimum length of the wire?
- Pedro places the top of his 6-metre ladder so that it comes directly to the top of a wall. How high is the wall if the bottom of the ladder is 3 metres from the bottom of the wall?
- Maria is holding her kite string that is 18 metres above the ground. If the string is 23 metres long, how far is a point directly under the kite?