To determine which statements regarding the proof of the Pythagorean Theorem are true, let's analyze each option carefully:
-
Length of a + length of b = length of c
- False: This does not apply in the context of the Pythagorean theorem. The theorem states that \( a^2 + b^2 = c^2 \), not \( a + b = c \).
-
The yellow triangles have an area of a x b, which would = c
- False: This statement does not correctly represent the area of any triangle. The area of a triangle is given by \( \frac{1}{2} \times \text{base} \times \text{height} \), not simply \( a \times b \). Additionally, it is unclear why it would equal \( c \).
-
The four yellow triangles on the left all have the same area as the four yellow triangles on the right (a+b/2) - therefore, the two small green squares on the left have to have the same area as the large green square on the right.
- True: This reflects the idea that the areas of the triangles on both sides must balance, leading to the conclusion that the remaining areas (the green squares) are equal.
-
The area of both large squares are equal; so the triangles with the square(s) must have a total area that is equal to each other.
- True: This statement correctly summarizes a key aspect of the Pythagorean theorem proof, which shows that the area of the squares built on the two legs (a and b) together equals the area of the square built on the hypotenuse (c).
-
The length of a is the same as the length of c - (a+b).
- False: This is not correct based on the geometry of right triangles. The hypotenuse is longer than either leg of the triangle, and this equation does not make sense in the context of the theorem.
Therefore, the two statements that are true in this context are:
- The four yellow triangles on the left all have the same area as the four yellow triangles on the right (a+b/2) - therefore, the two small green squares on the left have to have the same area as the large green square on the right.
- The area of both large squares are equal; so the triangles with the square(s) must have a total area that is equal to each other.