Question

Two posts are \(25 \mathrm{~m}\) and \(15 \mathrm{~m}\) high and the line joining their tips makes an angle of \(45^{\circ}\) with horizontal. The distance between these posts is : (a) \(5 \mathrm{~m}\) (b) \(10 / \sqrt{2} \mathrm{~m}\) (c) \(10 \mathrm{~m}\) (d) \(10 \sqrt{2} \mathrm{~m}\)

Short answer

Answer: The distance between the two posts is \(10\sqrt{2} \mathrm{~m}\).

Step-by-step solution

Draw a diagram

Draw a diagram representing the two posts, their heights, the horizontal line, and the angle between this line and the line joining the tips of the posts.

Identify right triangles

Identify two right triangles: a "small" one formed by the tip of the \(15\mathrm{~m}\) post, the tip of the \(25\mathrm{~m}\) post, and the point on the horizontal line vertically aligned with the second post, and a "big" one formed by the point on the ground vertically aligned with the first post, the point on the ground vertically aligned with the second post, and the tip of the \(25\mathrm{~m}\) post.

Find the length of the hypotenuse of the small triangle

Using the Pythagorean theorem, find the length of the hypotenuse of the small right triangle, which is also a side of the big right triangle. Since it is a triangle with sides in ratio \(1:1:\sqrt{2}\), having one of the sides (\(a = 25\mathrm{~m} - 15\mathrm{~m} = 10\mathrm{~m}\)), we can find the length of the hypotenuse using the formula \(c = a*\sqrt{2} = 10\sqrt{2}\mathrm{~m}\).

Determine the distance between the posts

Now, we can use trigonometry to find the distance between the posts, which is the length of the base of the big right triangle. We have the adjacent side (\(10\sqrt{2}\mathrm{~m}\)) and angle \(45^{\circ}\); therefore, we can use the tangent function: \(\tan(45^{\circ}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{distance between posts}}{10\sqrt{2} \mathrm{~m}}\) Now, since \(\tan(45^{\circ})=1\), we can find the distance between the posts by multiplying the adjacent side by the tangent function: \(\text{distance between posts} = 10\sqrt{2} \mathrm{~m} * 1 = 10\sqrt{2} \mathrm{~m}\). The correct answer is (d) \(10\sqrt{2} \mathrm{~m}\).

Key concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics, specifically within the subject of trigonometry, that applies to right-angled triangles.

It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship is often written as \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse while \( a \) and \( b \) are the other two sides.

When solving trigonometry problems involving right triangles, the Pythagorean theorem can be utilized to find a missing side length when the other two side lengths are known. It serves as a basic tool for establishing the relationships between the sides of a right triangle, and it's pivotal when linking angles with side lengths as seen in the given exercise.

Right Triangles

Right triangles are a special category of triangles that have one angle equal to \(90^{\text{o}}\). They are the cornerstone of trigonometry. The two sides adjacent to the right angle, often referred to as the 'legs', and the side opposite the right angle, known as the 'hypotenuse', have specific relationships that are consistent across all right triangles.

These relationships allow for the application of the Pythagorean theorem and also trigonometric ratios, such as the sine, cosine, and tangent functions, to solve for missing side lengths or angles. In problems involving right triangles, it is crucial to identify which sides correspond to the hypotenuse and the legs, in order to correctly apply the trigonometric principles.

The exercise showcased the utility of recognizing these elements within two nested right triangles and used this understanding to find a specific distance.

Tangent Function

The tangent function is one of the basic trigonometric functions, which relates the angles of a right triangle to the ratios of two of its sides. The tangent of an angle in a right triangle is defined as the ratio of the side opposite the angle to the side adjacent to it. This is mathematically represented as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

This function is especially useful when dealing with right triangles where one angle and one side are known, and there is a need to determine the other non-hypotenuse side. It was precisely used in the given problem, where the angle and the length of one side were known, and the tangent function provided the means to calculate the distance between the tops of the two posts. Knowing the specific value of the tangent function for common angles such as \(45^{\text{o}}\), where \( \tan(45^{\text{o}})=1 \), simplifies the calculation, allowing one to directly equate the opposite side to the adjacent side.