Question

A small commuter airline flies to three cities whose locations form the vertices of a right triangle (see figure). The total flight distance (from City A to City B to City \(C\) and back to City \(A\) ) is 1400 miles. It is 600 miles between the two cities that are farthest apart. Find the other two distances between cities.

Short answer

The other two distances between cities are 300 miles and 500 miles.

Step-by-step solution

Identify the Hypotenuse

Identify the longest distance between the cities, which is the hypotenuse of the right triangle. This equals to 600 miles.

Find the Sum of the Other Two Sides

Subtract the length of the hypotenuse from the total distance. The result, 1400 miles - 600 miles = 800 miles, is the sum of the distances between cities A and B, and cities A and \(C\). Let's denote these distances as x and y, respectively.

Set up the Pythagorean Theorem

According to the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the other two sides. Thus, we have \(x^2 + y^2 = 600^2 = 360000.\)

Write Down the Equation for the Sum of x and y

From our step 2, we know that \(x + y = 800\).

Solve the System of Equations

By squaring \(x + y = 800\), we get \(x^2 + y^2 + 2xy = 640000\). Comparing this equation with the equation from the Pythagorean theorem, we could subtract \(x^2 + y^2\) from both sides and get \(2xy = 640000 - 360000 = 280000\). By dividing each side by 2, we obtain \(xy = 140000\). Knowing that x and y add up to 800, we can solve for x and y using the quadratic equation \(x = (Sum - sqrt{Sum^2 - 4*Product})/2\) and \(y = (Sum + sqrt{Sum^2 - 4*Product})/2\). By substituting Sum = 800 and Product = 140000, we get \(x = 300\) and \(y = 500\).

Key concepts

Right Triangle

A right triangle is a type of triangle that has one angle of exactly 90 degrees.
This is known as the right angle. The longest side of a right triangle, opposite the right angle, is called the hypotenuse.
The other two sides are referred to as the legs. Understanding right triangles is critical when applying the Pythagorean Theorem. This theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this exercise, the vertices of the three cities form a right angle, allowing us to use the theorem to find the unknown distances. Remember, before using the Pythagorean Theorem:

• Ensure the triangle is a right triangle.
• Identify the hypotenuse, which is the longest side.

By applying these concepts, we can effectively solve for unknown distances in problems related to right triangles.

Distance Calculation

Calculating distances in a right triangle involves careful application of the Pythagorean Theorem.
In this exercise, you start by recognizing that the longest distance between cities is the hypotenuse. Given:

• Total distance of a round trip: 1400 miles
• Hypotenuse (longest side): 600 miles

The remaining two sides, representing the other distances, must sum up to 800 miles (1400 total - 600 hypotenuse).
Using these values, the Pythagorean Theorem helps us set up an equation.This involves:

• Calculating the sum of squares of the two unknown sides: \(x^2 + y^2\)
• Equating it to the square of the hypotenuse: \(600^2 = 360000\)

Distance calculations in such scenarios are foundational to solving more complex geometry problems. They illustrate how simple arithmetic and algebra can be used to determine unknown lengths.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In this problem, we use a system of equations to find distances between cities.
This is particularly useful when separate equations share variables that need to be solved together. In the airline problem, we had two key equations:

• The sum of the unknown city distances: \(x + y = 800\)
• The result from the Pythagorean Theorem: \(x^2 + y^2 = 360000\)

By solving this system:

• We first manipulate these equations to isolate one variable.
• Next, we use substitution or elimination to find exact values.

The quadratic equation derived from these relations lets us accurately calculate the distances \(x = 300\) miles and \(y = 500\) miles.
Mastering systems of equations enables us to handle multiple unknowns efficiently in diverse problem-solving scenarios.