Question

the corner of a room where two walls meet the floor should be a right triangle. jeff makes a mark along each wall. one mark is 3 inches from the corner. the other is 4 inches from the corner. how can jeff use the pythagorean theorem to see if the walls form a right angle?
if the distance across the floor from one mark to the other is \\(\square\\) inch(es), then the \\(\square\\) says that because \\(3^2 + 4^2 = \square^2\\), a right triangle is formed.

options: pythagorean theorem, converse of the pythagorean theorem

Explanation:

Step1: Recall the Converse of Pythagorean Theorem

The converse of the Pythagorean Theorem states that if \(a^2 + b^2 = c^2\) for a triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), then the triangle is a right triangle. Here, the two legs of the right triangle (if it is a right triangle) are \(a = 3\) inches and \(b = 4\) inches.

Step2: Calculate \(c\) using the Pythagorean Theorem

We know that for a right triangle, \(a^2 + b^2 = c^2\). Substituting \(a = 3\) and \(b = 4\), we get:
\[

$$\begin{align*} 3^2 + 4^2&= c^2\\ 9 + 16&= c^2\\ 25&= c^2 \end{align*}$$

\]
Taking the square root of both sides, we find \(c = \sqrt{25}= 5\) inches.

Step3: Determine the Theorem Used

Since we are using the fact that if \(3^2 + 4^2 = 5^2\), then the triangle is a right triangle, we are using the converse of the Pythagorean Theorem. The distance across the floor (the hypotenuse) should be \(5\) inches if a right angle is formed.

Answer:

The distance across the floor from one mark to the other is \(\boldsymbol{5}\) inches. The theorem used is the converse of the Pythagorean Theorem because \(3^2 + 4^2 = 5^2\), so a right triangle is formed.