Problem 25 - Intermediate Algebra

Question

Find the missing side lengths in each triangle. Give the exact answer and then an approximation to two decimal places when appropriate. See Example 2. One leg of an isosceles right triangle is 3.2 feet long. Find the length of its hypotenuse. Give the exact answer and then an approximation to two decimal places.

Step-by-Step Solution

Answer

Hypotenuse exact length: \(\sqrt{20.48}\) feet, approximate: 4.53 feet.

Step 1: Understand the properties of an isosceles right triangle

In an isosceles right triangle, the two legs are of equal length, and the hypotenuse is the longest side. The angles opposite the legs are both 45°. This means the triangle can be considered a 45°-45°-90° triangle.

Step 2: Apply the Pythagorean Theorem

In an isosceles right triangle, if each leg is of length \(a\), the hypotenuse \(c\) can be found using the Pythagorean theorem: \(c = \sqrt{a^2 + a^2} = \sqrt{2a^2}\).

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental concept in geometry, particularly when dealing with right-angled triangles. The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, which are referred to as the legs. The formula is expressed as: \[ c^2 = a^2 + b^2 \] where \(c\) represents the hypotenuse, while \(a\) and \(b\) are the legs of the triangle. In the context of an isosceles right triangle, this theorem helps us find the length of the hypotenuse when we know the length of the legs. When both legs are of equal length, this simplifies to the formula: \[ c = \sqrt{2a^2} \] Understanding this as the basis of our triangle calculations allows us to solve for unknown side lengths with ease. The theorem not only applies to isosceles right triangles but all right-angled triangles.

45°-45°-90° triangle

A 45°-45°-90° triangle is a special type of right triangle. As the name suggests, this triangle features two angles of 45 degrees and one right angle of 90 degrees. The unique property of this triangle is that it is always isosceles, meaning the two legs are of equal length. This makes calculations simpler and provides a consistent relationship between the side lengths:

• The two legs are equal (\(a = b\)
• The hypotenuse (\(c\)) can be determined directly from the legs using the Pythagorean theorem: \[ c = a\sqrt{2} \] This formula emerges because the sum of the squares of identical legs results in a simple relationship involving the square root of 2, a property unique to 45°-45°-90° triangles. Importantly, such triangles are prevalent in geometry problems that require precise and systematic calculations.

Hypotenuse calculation

Calculating the hypotenuse is a common task when dealing with right triangles. In the context of an isosceles right triangle with legs of length 3.2 feet, the hypotenuse \(c\) can be found using the derived formula for a 45°-45°-90° triangle: \[ c = a\sqrt{2} \] Substituting the given leg length, we have: \[ c = 3.2\sqrt{2} \] To find the exact value, it's crucial to understand the role of the square root of 2 as an irrational number, which leads to an approximation for practical purposes. By calculating \(3.2\sqrt{2}\) with a calculator, we approximate the hypotenuse length to two decimal places, providing a realistic measurement for real-world applications. This process illustrates how theoretical calculations translate into tangible dimensions and is especially useful in fields such as architecture, engineering, and everyday measurements.