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, [math]a^2+b^2=c^2[/math]. 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 [img]data:image/png;base64,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[/img] in quadrant I on the circumference of Circle O. Rename the point H by right clicking on the point and clicking on "rename".[br]Construct a perpendicular line [img]data:image/png;base64,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[/img] from point H to the x-axis. Create a point at the intersection of the perpendicular lines [img]data:image/png;base64,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[/img] and label it J.[code][/code][code][/code][code][/code][code][/code][code][/code][code][/code][code][/code][code][/code][code][/code][code][/code][br]Construct right triangle HJO [img]data:image/png;base64,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[/img].
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 [math]\angle HOJ[/math] [math]\theta[/math]. Define the length of the horizontal leg of triangle [math]HOJ[/math] in terms of [math]\theta[/math]. For a hint, type "Give me a hint!" and click "check my answer"
Task 5:
Define the length of the vertical leg of triangle [math]HOJ[/math] in terms of [math]\theta[/math]. 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 [math]HOJ[/math]? 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.
[br][br]Congratulations! You just found the Pythagorean Identity. It true for all values of [math]\theta[/math], and we can use it to solve various problems.
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Information: Pythagorean Identity - 2.6.5