Problem 42 - Elementary and Intermediate Algebra

Question

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) right triangle, the length of the hypotenuse is \(12 \sqrt{3}\) inches. Find the length of the leg opposite the \(30^{\circ}\) angle and the length of the leg opposite the \(60^{\circ}\) angle. Give the exact answer and then an approximation to two decimal places, when appropriate.

Step-by-Step Solution

Answer

Leg opposite 30°: \(6\sqrt{3}\) inches or 10.39 inches. Leg opposite 60°: 18 inches.

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

In a 30-60-90 triangle, the side opposite to the 30-degree angle is half the hypotenuse. The side opposite to the 60-degree angle is the length of the side opposite the 30-degree angle multiplied by \(\sqrt{3}\).

Step 2: Finding the Length of the Leg Opposite the 30-Degree Angle

Let the leg opposite the 30-degree angle be \(x\). For a 30-60-90 triangle, \(x = \frac{1}{2} \cdot \text{{hypotenuse}}\). Therefore, \(x = \frac{1}{2} \cdot 12\sqrt{3} = 6\sqrt{3}\) inches.

Step 3: Finding the Length of the Leg Opposite the 60-Degree Angle

Let the leg opposite the 60-degree angle be \(y\). For a 30-60-90 triangle, \(y = x \cdot \sqrt{3}\). Hence, \(y = 6\sqrt{3} \cdot \sqrt{3} = 18\) inches.

Step 4: Calculating Decimal Approximations

To provide decimal approximations, compute \(6\sqrt{3}\) and \(18\). Since \(\sqrt{3} \approx 1.732\), we find that \(6\sqrt{3} \approx 6 \times 1.732 = 10.392\) inches. The side \(y = 18\) is already an integer, so no approximation is needed.

Key Concepts

Understanding Right Triangle Properties

Right triangles are fundamental shapes in geometry, characterized by having one angle measuring exactly 90 degrees. This makes them particularly easy to work with in mathematics because specific relationships hold between their side lengths and angles.

In a 30-60-90 triangle, a special kind of right triangle, the properties are even more defined. These triangles have angles that measure exactly 30, 60, and 90 degrees. One of the key properties is that the hypotenuse is always twice as long as the shortest side, which is opposite the 30-degree angle.

• The side opposite the 30-degree angle (shorter leg) is half the hypotenuse.
• The side opposite the 60-degree angle (longer leg) is the shorter leg multiplied by the square root of 3.
• This setup allows for straightforward calculations and quick checks for accuracy when dealing with such triangles.

These consistent ratios make the 30-60-90 triangle predictable and easy to analyze with simple mathematical operations.

Calculating Triangle Side Lengths

To find the side lengths of a 30-60-90 triangle, you need to use its inherent properties and ratios. In our specific problem, the hypotenuse is given, and from it, you can deduce the lengths of the other sides.

Let's take this a step at a time:

• The hypotenuse is given as \(12\sqrt{3}\). For the side opposite the 30-degree angle, it's half of the hypotenuse. So, \(x = \frac{1}{2} \times 12\sqrt{3} = 6\sqrt{3}\).
• Next, the side opposite the 60-degree angle is the length opposite the 30-degree angle times \(\sqrt{3}\). So, \(y = 6\sqrt{3} \times \sqrt{3} = 18\).

Using these formulas helps in efficiently calculating missing lengths without starting from scratch each time. This established relationship simplifies solving geometric problems related to these specific triangle types.

Understanding Decimal Approximation

Decimal approximation involves converting precise mathematical results, often in terms of square roots or fractions, into a simpler decimal format for ease of understanding and practical use.

In this problem, we began with the side lengths expressed with square roots, specifically \(6\sqrt{3}\). It's beneficial to approximate this to better comprehend the real-world measurements.

• First, recognize that \(\sqrt{3} \approx 1.732\). By multiplying this with 6, calculate \(6 \times 1.732 = 10.392\).
• Thus, \(6\sqrt{3}\) approximately equals 10.392 inches. This provides a clear understanding of the physical length when square roots are less intuitive.
• The other leg, calculated as 18, requires no approximation since it's already in decimal form, making it straightforward.

Generally, having decimal approximations assists in practical scenarios, like when measuring or applying these calculations to real objects.