A right triangle is bordered on each side by a square. The area of Square 1
is A1,
the area of Square 2
is A2,
and the area of Square 3
isA3.
Which statement is true?
Responses
A1=A2+A3
cap A sub 1 is equal to cap A sub 2 plus cap A sub 3
A1=A2=A3
cap A sub 1 is equal to cap A sub 2 is equal to cap A sub 3
A1+A3=A2
cap A sub 1 plus cap A sub 3 is equal to cap A sub 2
A1+A2=A3
(cap A sub 1 plus cap A sub 2 is equal to cap A sub 3)
cap A sub 1 plus cap A sub 2 is equal to cap A sub 3
To determine which statement is true, we need to understand the relationship between the areas of the squares.
In this scenario, a right triangle is bordered by three squares. Let's label them as Square 1, Square 2, and Square 3.
Since the right triangle is bordered by each side of the square, we can imagine that the sides of the right triangle are the diagonals of the squares.
Now, let's consider the areas of the squares.
The area of a square is calculated by multiplying the length of one side by itself. Let's assume the lengths of the sides of Square 1, Square 2, and Square 3 are a, b, and c, respectively.
The area of Square 1 (A1) = a^2
The area of Square 2 (A2) = b^2
The area of Square 3 (A3) = c^2
Since the right triangle is bordered by these squares, we can say that:
a = b + c
Now, let's compare the statements given:
1. A1 = A2 + A3: This statement proposes that the area of Square 1 is equal to the sum of the areas of Square 2 and Square 3. Given that a^2 equals the sum of b^2 and c^2, this statement is True.
2. A1 = A2 = A3: This statement suggests that the three squares have the same area. However, since the right triangle is bordered by these squares, it is not possible for all three squares to be equal in size. Therefore, this statement is False.
3. A1 + A3 = A2: This statement implies that the sum of the areas of Square 1 and Square 3 is equal to the area of Square 2. This statement is not valid based on the equation a = b + c.
4. A1 + A2 = A3: This statement suggests that the sum of the areas of Square 1 and Square 2 is equal to the area of Square 3. Since we established that a^2 = b^2 + c^2, this statement is False.
Therefore, the statement that is True is: A1 = A2 + A3.