Question

A man travels on a bicycle, \(10 \mathrm{~km}\) east from the starting point \(\mathrm{A}\) to reach point \(\mathrm{B}\), then he cycles \(15 \mathrm{~km}\) south to reach point \(\mathrm{C}\). Find the shortest distance between \(\mathrm{A}\) and \(\mathrm{C}\). (1) \(25 \mathrm{~km}\) (2) \(5 \mathrm{~km}\) (3) \(25 \sqrt{13} \mathrm{~km}\) (4) \(5 \sqrt{13} \mathrm{~km}\)

Short answer

Answer: The shortest distance between points A and C is \(5 \sqrt{13} \mathrm{~km}\).

Step-by-step solution

Given information

The man cycles 10 km east (from point A to point B) and then 15 km south (from point B to point C). We need to find the shortest distance between points A and C.

Visualize the problem

Draw a right triangle with vertex A representing the starting point, vertex B representing the eastward distance of 10 km, and vertex C representing the end point. The distance from point A to point B is 10 km, and the distance from point B to point C is 15 km.

Apply the Pythagorean theorem

In a right triangle, the square of the length of the hypotenuse (the diagonal distance between points A and C in this case) is equal to the sum of the squares of the lengths of the other two sides. Let AC be the hypotenuse, AB be the eastward distance (10 km), and BC be the southward distance (15 km). Using the Pythagorean theorem: \(AC^2 = AB^2 + BC^2\)

Substitute given values

Replace AB with 10 km and BC with 15 km: \(AC^2 = (10 \mathrm{~km})^2 + (15 \mathrm{~km})^2\)

Solve for AC

Calculate the square of the lengths and add them: \(AC^2 = 100 \mathrm{~km}^2 + 225 \mathrm{~km}^2 = 325 \mathrm{~km}^2\) Now, take the square root of both sides to get the length of AC: \(AC = \sqrt{325 \mathrm{~km}^2} = 5 \sqrt{13} \mathrm{~km}\)

Answer

The shortest distance between points A and C is \(5 \sqrt{13} \mathrm{~km}\). Therefore, the correct answer is option (4) \(5 \sqrt{13} \mathrm{~km}\).

Key concepts

Distance Calculation

To find the shortest distance between two points, we often use the Pythagorean Theorem within a right triangle. In this case, the man cycles 10 km east and 15 km south. If we draw a right triangle, the legs of the triangle will be the eastward and southward distances. The hypotenuse of this triangle will be the shortest path between the starting and ending points.

The formula used for finding the shortest distance is the Pythagorean Theorem, which states that for a right triangle:

• The square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, this is written as \[AC^2 = AB^2 + BC^2\]where:

• \(AC\) is the hypotenuse or the shortest distance we want to find.
• \(AB\) is 10 km, the eastward distance.
• \(BC\) is 15 km, the southward distance.

Right Triangle

A right triangle is a triangle with one of its angles exactly equal to 90 degrees. This property creates a scenario where one side, known as the hypotenuse, is directly opposite the right angle. The other two sides are referred to as the legs of the triangle.

Right triangles are fundamental in geometry and trigonometry for calculating distances and angles. They are commonly used in various practical applications, from navigation and construction to theoretical mathematics. In our exercise,
- Point A is the vertex where the journey starts.
- Point B is directly east of point A.
- Point C is directly south of point B.
This setup forms a right triangle with points A, B, and C, allowing us to apply the Pythagorean Theorem effectively.

Hypotenuse

In any right triangle, the hypotenuse is the longest side. It is opposite the right angle and represents the shortest path between two points in a Euclidean plane. In the problem we've examined, the hypotenuse is the direct connection between the starting and ending points, A and C.

To find the hypotenuse, we use the Pythagorean Theorem. Substituting the given side lengths, we calculate as follows:

• First, calculate the squares of both legs: \(AB = 10 \text{ km}\) gives \( (10 \text{ km})^2 = 100 \text{ km}^2\) and \(BC = 15 \text{ km}\) gives \((15 \text{ km})^2 = 225 \text{ km}^2\).
• Add these squares: \(100 \text{ km}^2 + 225 \text{ km}^2 = 325 \text{ km}^2\).
• Finally, take the square root to find the hypotenuse: \( \sqrt{325 \text{ km}^2} = 5 \sqrt{13} \text{ km}\).

This result signifies the shortest possible distance the man could travel between points A and C.