Understanding the Ladder and Street Width Problem

This question involves a ladder leaning against walls on opposite sides of a street, with the base of the ladder staying in the same position. We are given the length of the ladder and the heights it reaches on each side. We need to find the total width of the street.

The situation on each side of the street forms a right-angled triangle. The ladder is the hypotenuse, the wall represents one leg (the height the ladder reaches), and the distance from the foot of the ladder to the base of the wall is the other leg (part of the street width). We can use the Pythagorean theorem to solve this.

The Pythagorean 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 (legs). Mathematically, if 'c' is the hypotenuse and 'a' and 'b' are the legs, the theorem is represented as:

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

Applying the Pythagorean Theorem to Calculate Distances

Let's consider the two scenarios described in the problem separately.

Scenario 1: Ladder on the first side of the street

• Length of the ladder (hypotenuse) = 13 m
• Height reached on the wall (one leg) = 12 m
• Distance from the foot of the ladder to the first wall (other leg) = \(d_1\)

Using the Pythagorean theorem:

\(d_1^2 + 12^2 = 13^2\)

\(d_1^2 + 144 = 169\)

Subtracting 144 from both sides:

\(d_1^2 = 169 - 144\)

\(d_1^2 = 25\)

Taking the square root of both sides:

\(d_1 = \sqrt{25}\)

\(d_1 = 5 \text{ m}\)

So, the distance from the foot of the ladder to the base of the wall on the first side is 5 meters.

Scenario 2: Ladder on the second side of the street

• Length of the ladder (hypotenuse) = 13 m
• Height reached on the wall (one leg) = 5 m
• Distance from the foot of the ladder to the second wall (other leg) = \(d_2\)

Using the Pythagorean theorem:

\(d_2^2 + 5^2 = 13^2\)

\(d_2^2 + 25 = 169\)

Subtracting 25 from both sides:

\(d_2^2 = 169 - 25\)

\(d_2^2 = 144\)

Taking the square root of both sides:

\(d_2 = \sqrt{144}\)

\(d_2 = 12 \text{ m}\)

So, the distance from the foot of the ladder to the base of the wall on the second side is 12 meters.

Calculating the Total Street Width

The foot of the ladder is at a single point between the two walls. The width of the street is the sum of the distances from this point to each wall.

Street Width = Distance to first wall (\(d_1\)) + Distance to second wall (\(d_2\))

Street Width = \(5 \text{ m} + 12 \text{ m}\)

Street Width = \(17 \text{ m}\)

Summary of Calculations

The width of the street is 17 meters.

Revision Table: Key Concepts

Additional Information: Pythagorean Triples

The numbers (5, 12, 13) form a well-known set of integers called a Pythagorean triple. A Pythagorean triple consists of three positive integers \(a, b, c\), such that \(a^2 + b^2 = c^2\). In this problem, we encountered two right triangles with sides (5, 12, 13). Knowing common Pythagorean triples can sometimes help solve problems like this more quickly, as you might recognize the side lengths directly.

Other common Pythagorean triples include (3, 4, 5), (8, 15, 17), and (7, 24, 25).