Which geometric construction can be used to prove the $\mathrm{Pythagorean} \mathrm{Theorem}$ by using the similarity of triangles?

• A. Bisect one of the acute angles of the right triangle and analyze the resulting segments.
• B. Draw an altitude from the right angle of a right triangle to the hypotenuse, creating two smaller right triangles within the original triangle.
• C. Inscribe a circle inside the right triangle and relate its radius to the triangle's sides.
• D. Construct a square on each side of a right triangle and compare their areas directly.

Answer: (B)

Verified step by step guidance

1. Identify the right triangle and focus on the right angle vertex where the two legs meet.
2. Draw an altitude (a perpendicular segment) from the right angle vertex down to the hypotenuse, effectively splitting the original right triangle into two smaller right triangles.
3. Recognize that these two smaller right triangles are similar to the original triangle and to each other by the Angle-Angle (AA) similarity criterion, since they share angles.
4. Use the properties of similar triangles to set up proportions relating the sides of the smaller triangles to the sides of the original triangle.
5. From these proportions, derive the relationship between the sides that leads to the Pythagorean Theorem, showing that the sum of the squares of the legs equals the square of the hypotenuse.