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 12 inches from the corner. The other is 9 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 __ inch(es), then the __ says that because 12² + 9² = __ a right triangle is formed.
To determine if the walls form a right angle using the Pythagorean Theorem, we can label the two marks Jeff makes along each wall as points A and B. The distance from the corner to mark A is 12 inches, and the distance from the corner to mark B is 9 inches.
According to the Pythagorean Theorem, for a right triangle, the following relationship holds:
[ c^2 = a^2 + b^2 ]
where ( c ) is the hypotenuse (the distance across the floor from point A to point B), and ( a ) and ( b ) are the lengths of the other two sides.
1. First, we calculate ( a^2 ) and ( b^2 ):
– ( a = 12 ), so ( a^2 = 12^2 = 144 )
– ( b = 9 ), so ( b^2 = 9^2 = 81 )
2. Next, we add these values together:
– ( a^2 + b^2 = 144 + 81 = 225 )
3. Now, we find ( c ) by taking the square root of 225:
– ( c = sqrt{225} = 15 )
Thus, the distance across the floor from one mark to the other is 15 inches. The Pythagorean Theorem says that