Sarah joined three squares at their vertices to create the figure shown in the diagram. She then covered each of the three squares with square centimeter tiles.

Question

The image illustrates the Pythagorean theorem, showing three squares connected at their vertices. Two smaller squares, labeled "Region P" and "Region R," are positioned adjacent to the legs of a right triangle. The third square, "Region S," is larger and positioned along the hypotenuse of the right triangle. A shaded right triangle is formed within the squares, connecting the vertices and visually representing the relationship a² + b² = c², where a and b are the sides of the right triangle (and sides of Region P and Region R), and c is the hypotenuse (and side of Region S). This diagram demonstrates that the sum of the areas of the squares on the legs of a right triangle (Region P and Region R) is equal to the area of the square on the hypotenuse (Region S).

Based on the information, which statement is true?
Responses
A The number of tiles needed to cover both Region P and Region R is greater than the number of tiles needed to cover Region S.The number of tiles needed to cover both Region P and Region R is greater than the number of tiles needed to cover Region S.
B The number of tiles needed to cover Region R is the same as the number of tiles needed to cover both Region P and Region S.The number of tiles needed to cover Region R is the same as the number of tiles needed to cover both Region P and Region S .
C The number of tiles needed to cover Region S is greater than the number of tiles needed to cover both Region P and Region R.The number of tiles needed to cover Region S is greater than the number of tiles needed to cover both Region P and Region R .
D The number of tiles needed to cover Region S is the same as the number of tiles needed to cover both Region P and Region R.

Answer 1

To evaluate which statement is true, let's first define the areas of each region based on the Pythagorean theorem:

• Let the lengths of the legs of the right triangle be \( a \) and \( b \).
• The area of Region P (the square on one leg) is \( a^2 \).
• The area of Region R (the square on the other leg) is \( b^2 \).
• The area of Region S (the square on the hypotenuse) is \( c^2 \), where \( c \) is the length of the hypotenuse.

According to the Pythagorean theorem, we know that:

\[ a^2 + b^2 = c^2 \]

Now let's analyze the options:

A. The number of tiles needed to cover both Region P and Region R is greater than the number of tiles needed to cover Region S.

This would mean \( a^2 + b^2 > c^2 \), which contradicts the Pythagorean theorem. Therefore, this statement is false.

B. The number of tiles needed to cover Region R is the same as the number of tiles needed to cover both Region P and Region S.

This would imply that \( b^2 = a^2 + c^2 \), which again contradicts the Pythagorean theorem. So, this statement is false.

C. The number of tiles needed to cover Region S is greater than the number of tiles needed to cover both Region P and Region R.

This would mean \( c^2 > a^2 + b^2 \), which contradicts the Pythagorean theorem. Therefore, this statement is false.

D. The number of tiles needed to cover Region S is the same as the number of tiles needed to cover both Region P and Region R.

This is equivalent to saying \( c^2 = a^2 + b^2 \), which directly follows from the Pythagorean theorem. Thus, this statement is true.

The correct choice is: D. The number of tiles needed to cover Region S is the same as the number of tiles needed to cover both Region P and Region R.