Problem 110 - Algebra and Trigonometry

Question

The size of a television screen refers to the length of its diagonal. If the length of an HDTV screen is 28 inches and its width is 15.7 inches, what is the size of the screen, to the nearest inch?

Step-by-Step Solution

Answer

The size of the TV screen is approximately 32 inches.

Step 1: Identify the given measures

The width of the TV is given as 15.7 inches and the length is given as 28 inches. These will act as the sides of a right triangle.

Step 2: Apply the Pythagorean Theorem

We know that \(a^2 + b^2 = c^2\), where a and b are sides of the triangle and c is the hypotenuse. Here, we can set a = 15.7 (the width of the TV) and b = 28 (the length of the TV). So, to find our hypotenuse c, we calculate \(c = \sqrt{{15.7}^2 + {28}^2}\).

Step 3: Compute and Round

Perform the calculations to find the value of c. Once you have the value for c, round it to the nearest whole number for the size of the screen.

Key Concepts

Understanding the Right Triangle in Geometry

The right triangle is one of the fundamental shapes in geometry. It is characterized by one angle measuring exactly 90 degrees, also known as a right angle. The sides forming this angle are referred to as the 'legs' of the triangle, while the side opposite the right angle is termed the 'hypotenuse', which is also the longest side of the triangle.

In the context of the given exercise, imagine the television as standing upright; the width and height represent the legs of a right triangle, and the screen diagonal is the hypotenuse. Understanding this setup is crucial as it leads us into utilizing the Pythagorean Theorem for calculating the screen size.

A robust grasp of right triangles is not only essential for solving many geometric problems but also for appreciating their numerous real-world applications, like determining the shortest path between two points in a city grid, or in this case, the actual screen size of a television.

How to Calculate the Hypotenuse

The hypotenuse calculation is a classic application of the Pythagorean Theorem, which is a cornerstone of geometry. According to the theorem, in a right triangle, the square of the length of the hypotenuse (\(c\text{, which is the diagonal in this case}\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\text{ and }b\text{, which are the width and height respectively}\)). The formula is expressed as \(a^2 + b^2 = c^2\).

Following this theorem for the TV screen in our exercise, we calculate the hypotenuse using the formula:
\[c = \[15.7^2 + 28^2\]^{1/2}\text{ inches}\text{.}\]
After performing the calculations, the resulting value gives us the diagonal of the television, which must then be rounded to the nearest whole number to answer the exercise question—thereby finding the TV screen size.

Geometry in Real Life

Geometry isn't just limited to the confines of textbooks and classrooms—it permeates through our daily lives in numerous ways. Let's dissect how understanding geometry, particularly the Pythagorean Theorem, can be immensely useful in real-life situations.

The exercise with the HDTV is a perfect example: consumers often need to know the screen size, which is the diagonal, to determine if the TV will fit in a desired space. Another instance is navigation; whether you're walking in a straight line from one corner of a park to another or a pilot is charting the shortest flight path, right triangle principles help in finding the shortest distance. Additionally, engineers and architects use these concepts when designing structures and tools. The principle is truly universal, assisting not just students but professionals across various fields in solving practical problems.

Recognizing the value of geometry, particularly the Pythagorean Theorem, can illuminate its prevalence in the objects and practices around us every day.