Problem 29 - Elementary and Intermediate Algebra

Question

Pythagorean Theorem Problems High-Ropes Adventures Courses. A builder of a high-ropes adventure course wants to secure a pole by attaching a support cable from the anchor stake 8 yards from its base to a point 6 yards up the pole. How long should the cable be? (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The cable should be 10 yards long.

Step 1: Understand the Problem

We need to find the length of the support cable that connects from the ground (anchor stake) to a point on the pole. The anchor stake is 8 yards from the base of the pole, and the cable connects to a point 6 yards up the pole.

Step 2: Identify the Right Triangle

A right triangle is formed by the pole, the ground, and the cable. The pole is vertical (6 yards up), the distance from the pole's base to the anchor is horizontal (8 yards), and these two form the legs of the triangle. The cable will be the hypotenuse.

Step 3: Apply the Pythagorean Theorem

In a right triangle, the Pythagorean Theorem states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): \[ c^2 = a^2 + b^2 \]. Here, \( a = 6 \text{ yards} \) and \( b = 8 \text{ yards} \).

Step 4: Calculate the Hypotenuse

Using the Pythagorean Theorem, we calculate: \[ c^2 = 6^2 + 8^2 = 36 + 64 = 100 \]. Hence, \( c = \sqrt{100} = 10 \).

Step 5: Interpret the Result

The length of the cable is the hypotenuse of the triangle, which we have found to be 10 yards. Therefore, the cable should be 10 yards long.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle measuring 90 degrees. Imagine the triangle formed by the builder’s setup. It consists of a vertical pole, a horizontal distance on the ground, and the cable as the diagonal. This 90-degree angle is crucial, as it allows us to use the Pythagorean Theorem to solve problems involving lengths and distances.

• The two sides forming the right angle, called the perpendicular sides, are referred to as the "legs" of the triangle.
• The side opposite the right angle, which is the longest side, is known as the hypotenuse.

When dealing with right triangles in practical situations, such as construction or navigation, understanding the role of the 90-degree angle can greatly simplify calculations.

Hypotenuse

The hypotenuse is a key component of any right triangle and is always opposite the 90-degree angle. It represents the longest side because of its diagonal nature compared to the two shorter, leg sides. In the high-ropes course problem, the cable acts as the hypotenuse.

• This is the side we often want to calculate, as it usually represents a direct path or a diagonal measure.
• To find the hypotenuse using the Pythagorean Theorem, simply take the square root of the sum of the squares of the two legs (perpendicular sides of the triangle).

For instance, with legs of 6 and 8 yards in our exercise, the hypotenuse, or cable length, is calculated as 10 yards. This demonstrates how the theorem helps in determining distances efficiently.

Right Triangle Legs

The right triangle legs are the two sides that form the 90-degree angle. These legs are straightforward and run either parallel or perpendicular to coordinate axes whenever right triangles are graphed or mapped.

• In practical terms, one of these legs is often vertical and the other horizontal, as seen in the exercise where one part of the leg was vertical up the pole and the other horizontal across the ground.
• Knowing the lengths of the legs allows you to apply the Pythagorean theorem to find the hypotenuse, providing a full understanding of the triangle's dimensions.

The role of these legs is foundational in calculating any unknown side or angle in right triangle problems, making them essential for understanding geometry and trigonometry in real-world applications.