Problem 34 - Elementary and Intermediate Algebra

Question

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the length of the shorter leg is \(5 \sqrt{2}\) inches. Find the length of the hypotenuse and the length of the longer leg. Give the exact answer and then an approximation to two decimal places.

Step-by-Step Solution

Answer

Hypotenuse: \(10\sqrt{2}\) inches (~14.14), longer leg: \(5\sqrt{6}\) inches (~12.25).

Step 1: Understanding Triangle Properties

In a 30-60-90 triangle, the sides have a specific ratio: the shorter leg (opposite 30°) is denoted as \(x\), the longer leg (opposite 60°) is \(x\sqrt{3}\), and the hypotenuse is \(2x\). In this case, the shorter leg is \(5\sqrt{2}\).

Step 2: Find Hypotenuse Length

Using the property of the 30-60-90 triangle, where the hypotenuse is \(2x\), substitute \(x = 5\sqrt{2}\). Therefore, the hypotenuse is \(2 \times 5\sqrt{2} = 10\sqrt{2}\) inches.

Step 3: Approximate Hypotenuse

To find the decimal approximation: \(10\sqrt{2} \approx 10 \times 1.414 = 14.14\) inches.

Step 4: Find Longer Leg Length

For the longer leg, use the formula \(x\sqrt{3}\) where \(x = 5\sqrt{2}\). The longer leg is \(5\sqrt{2} \times \sqrt{3} = 5\sqrt{6}\) inches.

Step 5: Approximate Longer Leg

Approximate \(5\sqrt{6}\) as follows: since \(\sqrt{6} \approx 2.449\), then \(5 \times 2.449 \approx 12.245\). The approximate length of the longer leg is 12.25 inches.

Key Concepts

Trigonometry Demystified

Trigonometry might sound like a complex term, but it's really about understanding triangles and their angles. In geometry, especially in right-angled triangles, trigonometry helps us find relationships between the angles and lengths of the sides. This branch of mathematics is essential because it allows us to solve problems involving triangles. The beauty of trigonometry is in how it uses simple functions like sine, cosine, and tangent to make complex calculations simple. In a right triangle, for example, the sine of an angle is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.

In the case of a 30-60-90 triangle, trigonometric concepts aren't directly applied in calculations but understanding these relations helps solidify our comprehension of how side lengths relate. Mastering these fundamentals of trigonometry can make analyzing and calculating right triangles much easier.

Understanding Triangle Ratios

Triangle ratios help us understand the relationships between the side lengths of a triangle. In a special type of triangle known as a 30-60-90 triangle, these ratios are especially important. The sides of a 30-60-90 triangle are always in a particular ratio: 1:√3:2.

• The shortest side, opposite the 30° angle, is represented as \( x \).
• The side opposite the 60° angle, which is longer, is \( x\sqrt{3} \).
• Finally, the hypotenuse, opposite the right angle (90°), is the longest side and is \( 2x \).

These fixed ratios allow you to easily calculate any side of the triangle if you know the length of one side. In our exercise, the shorter leg is \( 5\sqrt{2} \), so we can easily apply the ratios to find the lengths of the other sides by simply substituting this value for \( x \). It's a straightforward, logical way to work with special right triangles.

Applying the Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that describes the relationship between the sides of a right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.

While the solution to our 30-60-90 triangle problem doesn't directly use the Pythagorean Theorem, the theorem validates the relationships derived from the triangle's specific ratios. It serves as a backdrop, ensuring the accuracy of our calculated sides. For example, if we've calculated that \( a = 5\sqrt{2} \), \( b = 5\sqrt{6} \), and \( c = 10\sqrt{2} \), substituting these into the Pythagorean formula confirms that the triangle is consistent with these side lengths:

\[ (5\sqrt{2})^2 + (5\sqrt{6})^2 = (10\sqrt{2})^2 \]

Using the theorem is an excellent way to verify your work with geometrically-derived ratios, ensuring all our calculations are coherent and accurate. It’s like having a mathematical safety net.