Problem 33 - Elementary and Intermediate Algebra

Question

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the length of the leg opposite the \(30^{\circ}\) angle is \(75 \mathrm{cm} .\) Find the length of the leg opposite the \(60^{\circ}\) angle and the length of the hypotenuse. Give the exact answer and then an approximation to two decimal places, when appropriate.

Step-by-Step Solution

Answer

Exact sides: opposite 60° is \(75\sqrt{3}\) cm, hypotenuse is 150 cm. Approximate values: opposite 60° is 129.75 cm.

Step 1: Understanding the Properties of a 30-60-90 Triangle

In a 30-60-90 triangle, the sides are in a specific ratio: the length of the side opposite the 30° angle is \(x\), the side opposite the 60° angle is \(x \sqrt{3}\), and the hypotenuse is \(2x\).

Step 2: Identify the Given Side

We know the length of the side opposite the 30° angle is 75 cm. Therefore, we can denote \(x = 75\) cm.

Step 3: Determine the Side Opposite the 60° Angle

To find the length of the side opposite the 60° angle, use the ratio for a 30-60-90 triangle: \(x \sqrt{3}\). Substitute \(x = 75\) to get: \(75 \sqrt{3}\) cm.

Step 4: Multiply by the Square Root of 3

Calculate \(75 \sqrt{3}\): approximately, \(\sqrt{3} \approx 1.73\), so \(75 \times 1.73 \approx 129.75\) cm. The exact length is \(75 \sqrt{3}\) cm, and the approximate length is 129.75 cm.

Step 5: Determine the Length of the Hypotenuse

The hypotenuse is \(2x\). Substitute \(x = 75\) to find: \(2 \times 75 = 150\) cm.

Key Concepts

Right Triangle Properties

Right triangles are a special type of triangle that include one angle measuring exactly 90 degrees. This right angle determines several characteristics and sets the stage for various geometric principles. Essential properties of right triangles include:

• One angle is always 90 degrees.
• The side opposite this angle is called the hypotenuse, which is the longest side of the triangle.
• The remaining two sides, often referred to as "legs," are shorter than the hypotenuse and their lengths help determine the triangle’s specific class, such as a 30-60-90 triangle.

In a right triangle, knowledge of the specific relationships between angles and sides helps us to calculate unknown lengths easily. For a triangle to qualify as a right triangle, it must adhere to these basic properties and use them to solve various mathematical problems efficiently.

Triangle Side Ratios

Triangle side ratios are fundamental in understanding various geometrical relationships, particularly in special right triangles like the 30-60-90 triangle. This specific triangle is known for its unique side ratios:

• The side opposite the 30-degree angle is the shortest and designated as \(x\).
• The side opposite the 60-degree angle is longer and can be calculated as \(x \sqrt{3}\).
• The hypotenuse, always opposite the 90-degree angle, is the longest and equals \(2x\).

These ratios are not arbitrary; they arise from the intrinsic properties of equilateral triangles, which can be bisected to form 30-60-90 triangles. In practical terms, these side ratios mean that if you know the length of one side of a 30-60-90 triangle, you can effortlessly determine the lengths of the other two.
For example, if the side opposite the 30-degree angle is 75 cm, it establishes all other side lengths without the need for further angle measurement.

Hypotenuse Calculation

Calculating the hypotenuse in any right triangle is straightforward once you understand the properties involved. Specifically, in a 30-60-90 triangle, the process utilizes the known ratios.The hypotenuse is calculated using the formula \(2x\), where \(x\) represents the length of the side opposite the 30-degree angle. This formula directly derives from the inherent geometrical properties of the 30-60-90 triangle. If the side opposite the 30-degree angle is 75 cm:\[ 2x = 2 \times 75 = 150 \text{ cm} \]This calculation shows both the power and simplicity of utilizing geometric principles and established ratios.
Such calculations simplify many real-world applications, from architectural design to simple geometric problems, making them highly valuable in diverse fields.