Pythagorean Identity - 2.6.5

Let's Review Pythagorean Theorem

In a previous class you learned about the Pythagorean Theorem, which states that for any right triangle with legs a and b and hypotenuse c,

. If you need a refresher, watch the video below:

Task 1:

Write a mathematical statement about the relationship between the sides of the right triangle below:

Task 2:

Plot a point

in quadrant I on the circumference of Circle O. Rename the point H by right clicking on the point and clicking on "rename". Construct a perpendicular line

from point H to the x-axis. Create a point at the intersection of the perpendicular lines

and label it J. Construct right triangle HJO

.

Task 3:

Notice how you can move point H around in Quadrant I. What is the length of the hypotenuse, no matter where you drag point H? For a hint, type "Give me a hint!" and click "check my answer"

Task 4:

Let's call

. Define the length of the horizontal leg of triangle

in terms of

. For a hint, type "Give me a hint!" and click "check my answer"

Task 5:

Define the length of the vertical leg of triangle

in terms of

. For a hint, type "Give me a hint!" and click "check my answer"

Task 6:

Label the lengths you defined in tasks 3-5 on the triangle below:

Task 7:

Using what we know about Pythagorean Theorem, what can we say about the relationship between the side lengths of triangle

? For a hint, type "Give me a hint!" and click "check my answer"

Pause here: Check with your teacher to make sure you got the correct answer for Task 7.

Congratulations! You just found the Pythagorean Identity. It true for all values of

, and we can use it to solve various problems.