Understanding the Ladder Problem with Pythagoras

This problem involves a ladder leaning against a building, which forms a right-angled triangle. The ladder is the hypotenuse, the distance from the building base to the foot of the ladder is one leg (the base), and the height the ladder reaches on the building is the other leg (the height).

We can use the Pythagorean theorem to solve this problem. The theorem states that in a right-angled 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.

Mathematically, if 'c' is the hypotenuse and 'a' and 'b' are the other two sides, the Pythagorean theorem is:

\(a^2 + b^2 = c^2\)

Initial Position of the Ladder

Let's first analyse the initial position of the ladder. We are given:

• Length of the ladder (hypotenuse), \(c = 25\) m
• Initial distance of the foot from the building (base), \(b_1 = 7\) m

Let the initial height the ladder reaches on the building be \(h_1\). Using the Pythagorean theorem:

\(h_1^2 + b_1^2 = c^2\)

Substituting the given values:

\(h_1^2 + 7^2 = 25^2\)

\(h_1^2 + 49 = 625\)

Now, we solve for \(h_1^2\):

\(h_1^2 = 625 - 49\)

\(h_1^2 = 576\)

To find \(h_1\), we take the square root of 576:

\(h_1 = \sqrt{576}\)

\(h_1 = 24\) m

So, initially, the ladder reaches a height of 24 m on the building.

Position of the Ladder After Slipping

Now, the top of the ladder slips down by 4 m. This changes the height the ladder reaches on the building. The length of the ladder remains the same.

• Length of the ladder (hypotenuse), \(c = 25\) m (unchanged)
• The top slips by 4 m, so the new height \(h_2 = h_1 - 4\) m

Let's calculate the new height:

\(h_2 = 24 - 4\)

\(h_2 = 20\) m

Let the new distance of the foot from the building be \(b_2\). Using the Pythagorean theorem again for the new position:

\(h_2^2 + b_2^2 = c^2\)

Substituting the known values for the new position:

\(20^2 + b_2^2 = 25^2\)

\(400 + b_2^2 = 625\)

Now, we solve for \(b_2^2\):

\(b_2^2 = 625 - 400\)

\(b_2^2 = 225\)

To find \(b_2\), we take the square root of 225:

\(b_2 = \sqrt{225}\)

\(b_2 = 15\) m

So, after the top slips, the foot of the ladder is 15 m from the base of the building.

Calculating the Distance the Foot Slides

The question asks by how much distance the foot of the ladder will slide. This is the difference between the new distance from the building (\(b_2\)) and the initial distance from the building (\(b_1\)).

Distance slid = \(b_2 - b_1\)

Substituting the values:

Distance slid = \(15 - 7\)

Distance slid = \(8\) m

Therefore, the foot of the ladder slides by 8 m.

Summary of Ladder Movement

Here is a summary of the initial and final positions:

The increase in the distance from the building base is the distance the foot slid: \(15 \text{ m} - 7 \text{ m} = 8 \text{ m}\).

Revision Table: Pythagorean Theorem Concepts

Additional Information on Right Triangles

Right-angled triangles are fundamental in geometry and trigonometry. Understanding the Pythagorean theorem is crucial for solving many problems involving distances and lengths.

• Pythagorean triples are sets of three positive integers a, b, and c, such that \(a^2 + b^2 = c^2\). Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). In our problem, both the initial position (7, 24, 25) and the final position (15, 20, 25) form Pythagorean triples (though the second one is a multiple of (3, 4, 5) - (3*5, 4*5, 5*5)).
• The converse of the Pythagorean theorem is also true: if the sides of a triangle satisfy the equation \(a^2 + b^2 = c^2\), then the triangle is a right-angled triangle.
• Trigonometric functions (sine, cosine, tangent) are defined based on the ratios of the sides of a right-angled triangle.