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Triangle Side Lengths (Investigation)



  • To practice simplifying expressions with radicals
  • To practice estimating irrational numbers
  • To explore rational and irrational numbers with right triangles


  • Construction paper
  • Protractor with ruler
  • Calculator
  • Pencil


Work on your own or in pairs.

  1. On your piece of paper draw the right triangles whose legs have the lengths given. Once you draw the legs use a ruler to connect them and make the hypotenuse.
    • Leg 1: 1.5 units, Leg 2: 2.5 units
    • Leg 1: 3 units, Leg 2: 4 units
    • Leg 1: 3.25 units, Leg 2: 4.1 units
    • Leg 1: 5.3 units, Leg 2: 7.2 units
    • Leg 1: 4.4 units, Leg 2: 8.3 units
  2. For each of the triangles calculate the length of the hypotenuse using the following form of the Pythagorean Theorem:

\text{hypotenuse}^2 = (\text{Leg }1)^2 + (\text{Leg }2) ^2


Right triangles are triangles that have a right angle. Use the picture provided to guide your drawings.


  1. What is the exact length of the hypotenuse for each triangle in simplest form?
  2. Can any of the hypotenuse lengths be expressed without the radical?
  3. Convert each of the legs of the triangle to mixed numbers.
  4. Create a number line going from 0 to 10.
  5. On your number line indicate where you believe each of the hypotenuses lies.
  6. Find the decimal representation of each hypotenuse to the nearest hundredth. Use this to check if your placement was right on the number line.
  7. Are the lengths of the legs of the triangles rational or irrational numbers? Explain.
  8. Are the lengths of the hypotenuses of the triangles rational or irrational numbers? Explain.
  9. Based on your two previous answers what do you notice?
  10. Pick two new leg lengths and find the hypotenuse using the same technique. What do you notice?


  • Pick one of the triangles. If you wanted to increase the length of each of the legs by 20% what would be the new lengths of each of the sides? Draw it.
  • What do you notice happened to the length of the hypotenuse?

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