To determine which scenarios describe a right triangle, we can use the Pythagorean theorem, which states that in a right triangle with legs of lengths \( a \) and \( b \), and a hypotenuse of length \( c \), the following equation holds:
\[ c^2 = a^2 + b^2 \]
In each scenario, the ladder forms the hypotenuse, while the height of the ladder against the wall is one leg, and the distance from the wall is the other leg.
Let's analyze each situation:
A:
Ladder length (\(c\)) = 12 feet
Height (\(a\)) = 8 feet
Distance from wall (\(b\)) = 6 feet
Check the Pythagorean theorem:
\[ 12^2 ?= 8^2 + 6^2 \]
\[ 144 ?= 64 + 36 \]
\[ 144 = 100 \] (not true, so this is not a right triangle)
B:
Ladder length (\(c\)) = 10 feet
Height (\(a\)) = 6 feet
Distance from wall (\(b\)) = 8 feet
Check the Pythagorean theorem:
\[ 10^2 ?= 6^2 + 8^2 \]
\[ 100 ?= 36 + 64 \]
\[ 100 = 100 \] (true, so this is a right triangle)
C:
Ladder length (\(c\)) = 18 feet
Height (\(a\)) = 9 feet
Distance from wall (\(b\)) = 12 feet
Check the Pythagorean theorem:
\[ 18^2 ?= 9^2 + 12^2 \]
\[ 324 ?= 81 + 144 \]
\[ 324 = 225 \] (not true, so this is not a right triangle)
D:
Ladder length (\(c\)) = 15 feet
Height (\(a\)) = 12 feet
Distance from wall (\(b\)) = 9 feet
Check the Pythagorean theorem:
\[ 15^2 ?= 12^2 + 9^2 \]
\[ 225 ?= 144 + 81 \]
\[ 225 = 225 \] (true, so this is a right triangle)
Final Answers: The situations that describe a right triangle are:
- B
- D