Investigation 1: 45-45-90 triangles

 

  1. Consider a square.
    1. What are the measures of each angle?
    2. When you draw in a diagonal of a square, two triangles are formed. What are the measures of the angles of one of these triangles?
    3. The half-square triangle is better known as a "45-45-90" triangle. Why do you suppose it is called that?

  2. We give special attention to the 45-45-90 triangles. They have simple properties that we use in many applications.
    In the figures below, markings are used to give information. For each of the following triangles, explain how the markings indicate that the triangles is a 45-45-90 triangle.

Now that you have seen what a 45-45-90 triangle is, let's investigate how the sides are related.

  1. Consider these two triangles:
    1. Explain why they are both 45-45-90 triangles.
    2. Explain why they are similar.
    3. Using the Pythagorean Theorem, show that the length of the hypotenuse for the smaller triangle is .
    4. Using proportional reasoning, explain why the length of the hypotenuse of the larger triangle will be .
    5. Using the Pythagorean Theorem, show that the length of the hypotenuse for the larger triangle is .
    6. Recalling what we learned about simplifying radicals a couple units ago, explain why . (Hint: 18 = 9·2)

  2. We can see these special 45-45-90 triangles easily when we use dot paper. Take a look to the right.
    1. The 2 by 2 triangle has its hypotenuse labeled as .
      1. Explain why by using proportional reasoning.
      2. Explain why by using properties of radicals. (Hint: 8 = 4·2)
    2. The 4 by 4 triangle has no labels.
      1. Find the length of the hypotenuse using proportional reasoning.
      2. Find the length of the hypotenuse using the Pythagorean Theorem.
    3. How long is the hypotenuse of the 5 by 5 right triangle?

  3. Since all 45-45-90 triangles are similar, we can use proportional reasoning to quickly find the lengths of any 45-45-90 triangle.
    That is, all 45-45-90 triangles are magnified or shrunken versions of the triangle below.
    1. If you scaled up the triangle with a scale factor of 12, what will the new side lengths be? (Hint: leave your answer in radical form for quick work.)
    2. If you enlarged the triangle with a scale factor of 90, what will the new lengths be?
    3. If you shrunk the triangle with a scale factor of 1/2, what will the new lengths be?

  4. Consider the blue triangle below.
    1. Explain why the blue triangle is a 45-45-90 triangle. (The dots are on a perfect grid--evenly spaced.)
    2. Explain how the picture shows why the legs of the blue triangle will be long.
    3. If the hypotenuse of the blue triangle were 10 long instead of 6, how long would the legs be?
    4. How long would the legs be if the hypotenuse were 1,000?
    5. Describe a process to find the length of the legs for any hypotenuse length on a 45-45-90 triangle.

  5. Consider the purple 45-45-90 triangle below.
    1. How many of the length diagonals will it take to measure out one of the legs of the purple triangle?
    2. If the hypotenuse were 7 long instead of 5, how long would the legs be?
    3. If the hypotenuse were x long, how long would the legs be?

  6. Extending: Let's look carefully at the triangles in #6 and #7.

    Redrawing the blue triangle and the small 1 by 1 triangle, we get the following picture and proportion.

    Since x/1 is the same as x, we have the leg of the blue triangle is .
    Somehow this is the same as since it takes 3 of the lengths on the dot paper to make one of the legs of the blue triangle.
    To algebraically see why , click on this link.


    Consider the purple triangle again.
    1. Write a proportion and find the length of one of the legs of the purple triangle like we did above.
    2. Algebraically, mimic what was done in the part a to show .
    3. In part 7a, you should have answered "2 and 1/2" of the lengths. Explain how the fraction really is "2 and 1/2" of the lengths. (Hint: what is 5 divided by 2?)

  7. Summarize. So far we have seen two basic looks at the special 45-45-90 triangle. Here is what they've looked like.

    One has integer lengths for the legs and the other has integer lengths for the hypotenuse.
    Really they are still the same 45-45-90 triangle idea.
    1. Since the blue triangle clearly shows the hypotenuse is times the length of one of the legs, multiply by and see that it is 12.
    2. Describe in your own words how to find the length of any of the sides of a 45-45-90 triangle when you know either the leg or hypotenuse.