If you begin with a right triangle with sides a, b, and

c and decrease the right angle so it is an acute angle,
how does a2+b2compare with c2

Answer this question for the situation where you
increase the right angle so that it is obtuse.

To answer the first part of your question, let's begin with a right triangle with sides a, b, and c. According to the Pythagorean theorem, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be expressed as:

c^2 = a^2 + b^2

Now, if we decrease the right angle of the right triangle to make it an acute angle, the lengths of sides a and b remain the same, but the length of the hypotenuse, c, will be greater than before. This means that when you square the lengths of a and b and add them together, the sum will be less than the square of the new longer hypotenuse.

So, for the situation where you decrease the right angle to an acute angle, a^2 + b^2 is less than c^2.

Moving on to the second part of your question, if you increase the right angle of the right triangle to make it an obtuse angle, the lengths of sides a and b will still be the same. However, the length of the hypotenuse, c, will now be smaller than before. This means that when you square the lengths of a and b and add them together, the sum will be greater than the square of this shorter hypotenuse.

So, for the situation where you increase the right angle to an obtuse angle, a^2 + b^2 is greater than c^2.

In summary, when you decrease the right angle, a^2 + b^2 is less than c^2, and when you increase the right angle, a^2 + b^2 is greater than c^2.