Question
A contractor needs to pour a concrete floor in the shape of a rectangle. The figure shows the design for the concrete floor. the top and bottom is 40 and the two sides is 30.The contractor uses the converse of the Pythagorean theorem to prove that the floor is rectangular. How can he do this?
A.
He can show that 40−30
equals the square of the length of each diagonal.
B.
He can show that 302+402
equals the square of the length of each diagonal.
C.
He can show that 402−302
equals the square of the length of each diagonal.
D.
He can show that 30+40
equals the square of the length of each diagonal.
A.
He can show that 40−30
equals the square of the length of each diagonal.
B.
He can show that 302+402
equals the square of the length of each diagonal.
C.
He can show that 402−302
equals the square of the length of each diagonal.
D.
He can show that 30+40
equals the square of the length of each diagonal.
Answers
GPT-4o mini
To prove that a shape is a rectangle using the converse of the Pythagorean theorem, the contractor needs to demonstrate that the sum of the squares of the two shorter sides is equal to the square of the length of the diagonal.
In this case, the sides of the rectangle are 30 (height) and 40 (width). According to the Pythagorean theorem, in a right triangle formed by the two sides and the diagonal, we have:
\(a^2 + b^2 = c^2\)
where:
- \(a\) = 30 (one side)
- \(b\) = 40 (other side)
- \(c\) = length of the diagonal
The converse of the Pythagorean theorem states that if \(a^2 + b^2 = c^2\) holds true, then the figure is a right triangle, and hence the rectangle formed by these sides is indeed a right rectangle.
Calculating:
1. \(30^2 = 900\)
2. \(40^2 = 1600\)
3. So, \(30^2 + 40^2 = 900 + 1600 = 2500\)
4. Therefore, \(c^2 = 2500\), implying \(c = \sqrt{2500} = 50\).
This proves that the relationship holds true for the diagonal.
The correct choice is therefore:
**B. He can show that \(30^2 + 40^2\) equals the square of the length of each diagonal.**
In this case, the sides of the rectangle are 30 (height) and 40 (width). According to the Pythagorean theorem, in a right triangle formed by the two sides and the diagonal, we have:
\(a^2 + b^2 = c^2\)
where:
- \(a\) = 30 (one side)
- \(b\) = 40 (other side)
- \(c\) = length of the diagonal
The converse of the Pythagorean theorem states that if \(a^2 + b^2 = c^2\) holds true, then the figure is a right triangle, and hence the rectangle formed by these sides is indeed a right rectangle.
Calculating:
1. \(30^2 = 900\)
2. \(40^2 = 1600\)
3. So, \(30^2 + 40^2 = 900 + 1600 = 2500\)
4. Therefore, \(c^2 = 2500\), implying \(c = \sqrt{2500} = 50\).
This proves that the relationship holds true for the diagonal.
The correct choice is therefore:
**B. He can show that \(30^2 + 40^2\) equals the square of the length of each diagonal.**