Final answer:
In this proof, we use the Pythagorean theorem to show that the sum of the squares of the lengths of the legs in a right triangle is equal to the square of the length of the hypotenuse.
Step-by-step explanation:
In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. This is known as the Pythagorean theorem. In the given proof, we have a right triangle ABC with the altitude CD drawn from the right angle. We need to prove that AC^2 + BC^2 = AB^2.
To prove this, we can use the Pythagorean theorem twice. First, in triangle ADC, we have AD^2 + CD^2 = AC^2. Second, in triangle BDC, we have BD^2 + CD^2 = BC^2. Adding both equations, we get AD^2 + BD^2 + 2CD^2 = AC^2 + BC^2.
But, AD^2 + BD^2 = AB^2 (by the Pythagorean theorem in triangle ABD). So, substituting this into the previous equation, we have AB^2 + 2CD^2 = AC^2 + BC^2. Since CD^2 = AB^2 (since CD is an altitude of the right triangle), we can substitute this into the equation to get AB^2 + 2AB^2 = AC^2 + BC^2. Simplifying, we obtain AC^2 + BC^2 = AB^2.