Problem 160 - Elementary Algebra

Question

A city planner wants to build a bridge across a lake in a park. To find the length of the bridge, he makes a right triangle with one leg and the hypotenuse on land and the bridge as the other leg. The length of the hypotenuse is 340 feet and the leg is 160 feet. Find the length of the bridge.

Step-by-Step Solution

Answer

The length of the bridge is 300 feet.

Step 1: Understand the Problem

We need to find the length of the bridge in a right triangle where the hypotenuse and one leg of the triangle are given. The hypotenuse is 340 feet, and one leg is 160 feet.

Step 2: Identify the Formula

The Pythagorean theorem relates the lengths of the sides of a right triangle: We have one leg ( The formula is: We are given Find

Step 3: Set Up the Equation

Plug in the known values of the hypotenuse and one leg into the Pythagorean theorem:

Step 4: Calculate the Length of the Bridge

Solve for both sides:

Step 5: Find the Square Root

Find the square root of both sides to get the length of the bridge:

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle equal to 90 degrees. This 90-degree angle is also called a right angle. In a right triangle, the side opposite to the right angle is the longest side and is known as the hypotenuse. The other two sides are referred to as the legs of the triangle.
Right triangles have special properties that make them unique:

• The sum of the squares of the legs is equal to the square of the hypotenuse.
• They are often used in various practical applications, including construction and navigation.

In this particular exercise, the city planner uses a right triangle to determine the bridge length.

Hypotenuse

The hypotenuse is the longest side of a right triangle, and it is always located opposite the right angle. The Pythagorean Theorem prominently involves the hypotenuse. The theorem states that for any right triangle:
\[a^2 + b^2 = c^2\text{, where }a\text{ and }b\text{ are the legs and }c\text{ is the hypotenuse.}\]
In our exercise, the hypotenuse is given as 340 feet. Knowing this, we can use its value along with one of the legs to find the other leg, which represents the length of the bridge.

Leg Length

The legs of a right triangle are the two sides that form the right angle. In the context of our problem, we know one of the legs is 160 feet long. To find the length of the other leg (the bridge), you can use the Pythagorean Theorem as follows:

• Plug in the known values: \[a^2 + (160)^2 = (340)^2\]
• Calculate the squares of the known values: \[a^2 + 25600 = 115600\]
• Simplify to find \(a^2\): \[a^2 = 115600 - 25600 = 90000\]
• Find \(a\) by taking the square root: \[a = \sqrt{90000} = 300 \text{ feet}\]

Thus, the leg length representing the bridge is 300 feet.

Bridge Length Calculation

To find the length of the bridge, we use the known values of the hypotenuse and one leg in the right triangle. Follow these steps:
1. Start with the Pythagorean Theorem: \[ a^2 + b^2 = c^2 \]2. Substitute the known values: \[ a^2 + 160^2 = 340^2 \]3. Compute the squares: \[ a^2 + 25600 = 115600 \]4. Solve for \(a^2\): \[ a^2 = 115600 - 25600 = 90000 \]5. Find the square root: \[ a = \sqrt{90000} = 300 \text{ feet} \]Thus, by employing the Pythagorean Theorem, you can calculate that the bridge must be 300 feet long.