Learning Pythagorean Theorem
Pythagorean Theorem:
Pythagorean theorem states that In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides. In order for this to hold , the triangle be a right triangle. This is represented by the formula a2+b2=c2, where a and b are the sides adjacent to the right angle, and c is the hypotenuse.
Visual of Pythagorean Theorem
Is the triangle ABC depicted above a right triangle?
Do non-right triangles have a hypotenuse?
Pythagorean Theorem Practice:
In the below visual, We use the formula A2+b2=C2. In our triangle, let a=the segment AB. b=BC, and c=AC. Toggle with the options to show the calculations and values of the sides to get a better understanding of the theorem.
Interacting With Side Lengths:
Alter the side lengths by moving the point A and/or B to see how the values for A2 , B2 and C2 change
Use the side lengths given to evaluate the formula
Calculate the value of A2+B2, and the value of C2. Did you find that the formula holds?
Question:
Move the points A,B, and C to change the side lengths. Is there some arrangement of the side lengths that invalidates the pythagorean theorem? Why or why not?
Is this triangle a right Angle?
Use the pythagorean theorem for the triangle ABC
Here we see when we look at triangle ABC, we could assume it's not a right triangle. However, when we evaluate a2+b2=c2, we see (3)2+(4)2=(5)2
9 + 16 = 25
25 =25 , here the triangle satisfies the pythagorean theorem, therefore triangle ABC IS a right triangle, although it may not appear like it. This is something your future teachers may try to fool you, remember that if a2+b2=c2 then the triangle is a right angle
Conclusion:
We have seen that regardless of the arrangement of the points A,B,C, and the varying side lengths, that the pythagorean theorem always holds. This is a useful theorem to use moving forward this will be used in various levels of mathematical courses. We can also use this theorem to prove that triangles are right triangles, if a2+b2=c2, even if they're visualized incorrectly. If the triangle satisfies our formula, it is always a right triangle.