Question

pythagorean proof
what is true about the area of the large square a and the smaller shapes inside?
pythagorean theorem
a = a1×a2×a3×a4×a5 a = a1+a2+a3+a4+a5
a = a1−a2−a3−a4−a5 a+a1 = a2+a3+a4+a5

Explanation:

Step1: Recall Area Addition Principle

The area of a composite figure (like the large square A) is the sum of the areas of its non - overlapping component parts (A1, A2, A3, A4, A5 here). Multiplication, subtraction, or the last option don't follow the basic area - composition rule.

Step2: Analyze Each Option

• For \(A = A1\times A2\times A3\times A4\times A5\): This is incorrect as area of a composite figure is not the product of the areas of its parts.
• For \(A = A1 + A2+A3 + A4+A5\): This follows the area addition postulate, as the large square is made up of the four right - angled triangles (A1, A2, A3, A4) and the inner square (A5), and the total area should be the sum of these individual areas.
• For \(A = A1 - A2 - A3 - A4 - A5\): This is incorrect as the area of the whole should be more than the parts, not less by subtracting the parts.
• For \(A + A1=A2 + A3+A4 + A5\): This is incorrect as it doesn't follow the area - composition logic.

Answer:

\(A = A1+A2+A3+A4+A5\) (the second option among the given choices)