Asked by nagumo

Choose all situations that describe a right triangle.
Responses
A A ladder 12 feet long leans against a wall. The top of the ladder is 8 feet above the ground. The bottom of the ladder is 6 feet from the wall.A ladder 12 feet long leans against a wall. The top of the ladder is 8 feet above the ground. The bottom of the ladder is 6 feet from the wall.
B A ladder 10 feet long leans against a wall. The top of the ladder is 6 feet above the ground. The bottom of the ladder is 8 feet from the wall.A ladder 10 feet long leans against a wall. The top of the ladder is 6 feet above the ground. The bottom of the ladder is 8 feet from the wall.
C A ladder 18 feet long leans against a wall. The top of the ladder is 9 feet above the ground. The bottom of the ladder is 12 feet from the wall.A ladder 18 feet long leans against a wall. The top of the ladder is 9 feet above the ground. The bottom of the ladder is 12 feet from the wall.
D A ladder 15 feet long leans against a wall. The top of the ladder is 12 feet above the ground. The bottom of the ladder is 9 feet from the wall.

Answers

Answered by GPT-4o mini
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**)

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**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**)

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**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**)

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**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**)

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**Final Answers:**
The situations that describe a right triangle are:
- B
- D
There are no human answers yet. A form for humans to post answers is coming very soon!

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