Geometry NameSpecial Right Triangles Investigation DateDay _____ CW Block

Knowing about patterns of special right triangles is useful because they can help to solve some problems quickly, sometimes even in your head.

1) Use the Pythagorean Theorem to find the length of the hypotenuse for the three isosceles (45-45-90) right triangles. Write the length in simplified radical form, NOT as a decimal approximation.

2) Look for a pattern between the leg lengths and the hypotenuse. Using your observations, sketch the next two figures in the sequence of isosceles right triangles.

figure 4 figure 5

3) Use your results for the first several triangles to express the length of c in terms of n. Write a sentence that tells how to find the length of the hypotenuse of any isosceles right triangle without using the Pythagorean Theorem.

4. Use the Pythagorean Theorem to find the length of the missing side for the 30-60-90 right triangles. Write the length in simplified radical form, NOT as a decimal approximation.

5. Look for a pattern between the leg lengths and the hypotenuse. Using your observations, sketch the next two figures in the sequence of isosceles right triangles.

figure 4 figure 5

6. Use your results for the first several triangles to express the length of the leg opposite of the 60° angle in terms of n. Write a sentence that tells how to find the length of the hypotenuse of any 30-60-90 right triangle without using the Pythagorean Theorem.

7. Use your observations about the patterns of 45-45-90 and 30-60-90 triangles to find the missing sides of the following.

c) d)