Finding the Distance Between the Tops of Two Poles

This problem involves two vertical poles of different heights standing on level ground. We are given the heights of the poles and the distance between their feet. We need to find the distance between their tops. This situation can be visualized as a right-angled triangle where the distance between the tops is the hypotenuse.

Understanding the Pole Problem Geometry

Let's represent the two poles as vertical lines. Pole 1 has a height of 15 m, and Pole 2 has a height of 20 m. They are standing on a plane ground, and the horizontal distance between their feet is 12 m.

To find the distance between their tops, we can draw a line segment connecting the tops of the two poles. Now, imagine drawing a horizontal line from the top of the shorter pole to the vertical line of the taller pole. This creates a right-angled triangle.

• The base of this right-angled triangle is the horizontal distance between the feet of the poles, which is 12 m.
• The vertical side of this right-angled triangle is the difference in height between the two poles.

Calculating the Difference in Height

The height of Pole 1 is \(h_1 = 15\) m.

The height of Pole 2 is \(h_2 = 20\) m.

The difference in height is \( \Delta h = h_2 - h_1 = 20 \text{ m} - 15 \text{ m} = 5 \text{ m} \).

This difference in height, 5 m, is one leg of our right-angled triangle.

Applying the Pythagorean Theorem

We have a right-angled triangle with:

• One leg (horizontal distance) = 12 m
• The other leg (vertical difference in height) = 5 m
• The hypotenuse (distance between the tops) = let's call it \(d_{tops}\)

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

The formula is \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse.

In our case, \(a = 12\) m and \(b = 5\) m. We want to find \(c = d_{tops}\).

So, we have:

\( d_{tops}^2 = (\text{distance between feet})^2 + (\text{difference in height})^2 \)

\( d_{tops}^2 = (12 \text{ m})^2 + (5 \text{ m})^2 \)

\( d_{tops}^2 = 144 \text{ m}^2 + 25 \text{ m}^2 \)

\( d_{tops}^2 = 169 \text{ m}^2 \)

To find \(d_{tops}\), we take the square root of 169:

\( d_{tops} = \sqrt{169 \text{ m}^2} \)

\( d_{tops} = 13 \text{ m} \)

Conclusion on Distance Calculation

The distance between the tops of the two poles is 13 m.

Revision Table: Key Concepts for Pole Problems

Additional Information: Extending the Concept

The same principle can be applied to similar problems involving vertical heights and horizontal distances. For example:

• Finding the length of a ladder leaning against a wall (forms a right triangle).
• Calculating the distance between two objects if their altitudes and horizontal separation are known.
• Problems involving shadows cast by vertical objects.

In all these cases, identifying or constructing a right-angled triangle is key to using the Pythagorean theorem effectively to find unknown lengths.