6. Sasha is building a tree house. The walls are 6.5 feet tall and she is using a brace to hold up the wall while she nails it to the foor, The brace is 8 feet long and she has positioned it 5 feet from the wall. Does her wall meet the floor at a right angle? Explain.

To determine if Sasha's wall meets the floor at a right angle, we can analyze the situation using the properties of a right triangle. ### Known Conditions: 1. The height of the wall (vertical side) = 6.5 feet. 2. The length of the brace (hypotenuse) = 8 feet. 3. The distance from the wall to the base of the brace (horizontal side) = 5 feet. ### Step-by-Step Solution: 1. **Identify the Triangle**: - The wall forms a vertical side of the triangle. - The distance from the wall to the base of the brace forms the horizontal side. - The brace itself is the hypotenuse. 2. **Use the Pythagorean Theorem**: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be expressed as: \[ c^2 = a^2 + b^2 \] Where: - \( c \) = length of the brace (8 feet) - \( a \) = height of the wall (6.5 feet) - \( b \) = distance from the wall to the base of the brace (5 feet) 3. **Substitute the Values**: \[ 8^2 = 6.5^2 + 5^2 \] 4. **Calculate Each Side**: - Calculate \( 8^2 \): \[ 8^2 = 64 \] - Calculate \( 6.5^2 \): \[ 6.5^2 = 42.25 \] - Calculate \( 5^2 \): \[ 5^2 = 25 \] 5. **Add the Squares of the Other Two Sides**: \[ 42.25 + 25 = 67.25 \] 6. **Compare the Results**: Now we compare \( 64 \) (from the hypotenuse) with \( 67.25 \) (from the other two sides): \[ 64 \neq 67.25 \] ### Conclusion: Since \( 64 \) is not equal to \( 67.25 \), the wall does not meet the floor at a right angle. The brace is not positioned correctly to support the wall at a right angle to the floor.