Which two points of concurrency always remain inside the triangle, and why?
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Answer:
See below.
Step-by-step explanation:
The converse of the Pythagorean Theorem tells us that if the sum of the square of the two shorter lengths is equivalent to the square of the longest length, then the triangle is indeed a right triangle.
We have the side lengths (x²-1), (2x), and (x²+1).
We can see that our hypotenuse will be (x²+1), since it will always yield the largest value for x.
So, we can use the Pythagorean Theorem:
Substitute (x²-1) and (2x) for a and b, and substitute (x²+1) for c:
Square each expression:
Combine like terms:
Add:
We can see that the resulting equations are equivalent, proving that the triangle is indeed a right triangle.
And we're done!