A ladder 6.5 m long is standing against a wall and the difference between the base of the ladder and wall is 5.2 m. If the top of the ladder now slips by 1.4 m, then by how much will the foot of the ladder slip?

Options

A. 1.2 m
B. 0.8 m
C. 0.75
D. can't be determined

Answer

To solve the problem step by step, we will use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

### Step 1: Identify the lengths
We have a ladder (AC) that is 6.5 m long, and the distance from the wall to the base of the ladder (BC) is 5.2 m. We need to find the height of the ladder on the wall (AB) using the Pythagorean theorem.

### Step 2: Apply the Pythagorean theorem to triangle ACB
Using the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ 6.5^2 = AB^2 + 5.2^2 \]
Calculating the squares:
\[ 42.25 = AB^2 + 27.04 \]
Now, isolate \(AB^2\):
\[ AB^2 = 42.25 - 27.04 = 15.21 \]
Taking the square root to find \(AB\):
\[ AB = \sqrt{15.21} \approx 3.9 \text{ m} \]

### Step 3: Determine the new height after the slip
The top of the ladder slips down by 1.4 m, so the new height (AE) is:
\[ AE = AB - 1.4 = 3.9 - 1.4 = 2.5 \text{ m} \]

### Step 4: Apply the Pythagorean theorem to triangle DBE
Now we will use the Pythagorean theorem again to find the new distance from the wall to the base of the ladder (DC). The new triangle DBE has:
\[ DE^2 = BE^2 + DC^2 \]
Where \(DE\) (the length of the ladder) is still 6.5 m and \(BE\) (the new height) is 2.5 m.
Substituting the known values:
\[ 6.5^2 = 2.5^2 + DC^2 \]
Calculating the squares:
\[ 42.25 = 6.25 + DC^2 \]
Now, isolate \(DC^2\):
\[ DC^2 = 42.25 - 6.25 = 36 \]
Taking the square root to find \(DC\):
\[ DC = \sqrt{36} = 6 \text{ m} \]

### Step 5: Calculate the distance the foot of the ladder slips
The original distance from the wall to the base of the ladder (BC) was 5.2 m. The new distance (DC) is 6 m. Therefore, the foot of the ladder slips by:
\[ \text{Slip} = DC - BC = 6 - 5.2 = 0.8 \text{ m} \]

### Final Answer
The foot of the ladder slips by **0.8 meters**.

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