Problem 38 - Elementary and Intermediate Algebra

Question

In a \(30^{\circ}\) \(-60^{\circ}-90^{\circ}\) right triangle, the length of the longer leg is 24 yards Find the length of the hypotenuse and the length of the shorter leg. Give the exact answer and then an approximation to two decimal places.

Step-by-Step Solution

Answer

The shorter leg is \(8\sqrt{3}\) yards (\(\approx 13.86\) yards) and the hypotenuse is \(16\sqrt{3}\) yards (\(\approx 27.71\) yards).

Step 1: Understand the Properties of the Triangle

A \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle has specific properties: the length of the hypotenuse is twice the length of the shorter leg, and the longer leg is \( \sqrt{3} \) times the shorter leg.

Step 2: Identify Given Information

We are given that the longer leg of the triangle is 24 yards. Let the shorter leg be \( x \). Therefore, we can write an equation using the relationship of the sides: \( 24 = x \sqrt{3} \).

Step 3: Solve for the Shorter Leg

Rearrange the equation from Step 2 to solve for \( x \):\[x = \frac{24}{\sqrt{3}}.\]

Step 4: Rationalize the Denominator

To simplify \( \frac{24}{\sqrt{3}} \), multiply numerator and denominator by \( \sqrt{3} \):\[x = \frac{24 \cdot \sqrt{3}}{3}.\]which simplifies to \( x = 8\sqrt{3} \) yards.

Step 5: Find the Hypotenuse

Using the property that the hypotenuse is twice the shorter leg, calculate the hypotenuse:\[h = 2 \times 8\sqrt{3} = 16\sqrt{3} \text{ yards}.\]

Step 6: Approximate the Sides to Two Decimal Places

Calculate the decimal approximation:- For the shorter leg: \( 8\sqrt{3} \approx 13.86 \) yards,- For the hypotenuse: \( 16\sqrt{3} \approx 27.71 \) yards.

Key Concepts

Right Triangle Properties

In geometry, right triangles have a special place due to their unique properties. A right triangle always has one angle of 90 degrees. In a right triangle like the 30-60-90 triangle, the other two angles are 30 and 60 degrees respectively. Each angle in a triangle has a direct influence on the length of the sides opposite to it. This specific triangle is notable because it has a predictable ratio of side lengths:

• The side opposite the 30-degree angle (shorter leg) is the smallest.
• The hypotenuse is the longest side and is opposite the 90-degree angle.
• The side opposite the 60-degree angle is the longer leg.

The relationship between the sides of a 30-60-90 triangle is consistent. The hypotenuse is double the length of the shorter leg, while the longer leg is \(\sqrt{3}\) times the length of the shorter leg. This predictable ratio allows for quick calculations when solving for side lengths.

Trigonometry in Triangles

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. In a 30-60-90 right triangle, trigonometric ratios can be particularly useful. If you know one side, you can easily find the others using these ratios.

In our case, we are aware of the longer leg and want to determine the lengths of the other sides. Using trigonometric relationships, we set up the equation for the sides:

• The relationship for the longer leg is expressed as \(\text{longer leg} = \text{shorter leg} \times \sqrt{3}\).
• The hypotenuse-to-shorter-leg relation is often solved with \(\text{hypotenuse} = \text{shorter leg} \times 2\).

The use of trigonometric concepts and this special ratio makes solving the triangle a systematic and straightforward process.

Solving Triangles

Solving triangles involves finding unknown side lengths or angles. In the context of a 30-60-90 triangle, this becomes more of a pattern recognition exercise. Since each side is connected by a specific ratio, knowing one side allows you to solve for the others quickly.

Given the longer leg is 24 yards, we employ the earlier mention equation \(\text{longer leg} = \text{shorter leg} \times \sqrt{3}\) and solve for the shorter leg:
\[x = \frac{24}{\sqrt{3}}\]
To simplify to a radical form, multiply by \(\sqrt{3}/\sqrt{3}\) and obtain \(x = 8\sqrt{3}\) yards.

Next, calculate the hypotenuse using the simple relationship:
\[h = 2 \times 8\sqrt{3} = 16\sqrt{3}\] yards.
Approximations give us practical answers:

• The shorter leg: 8\(\sqrt{3}\) approximates to 13.86 yards.
• The hypotenuse: 16\(\sqrt{3}\) approximates to 27.71 yards.

The precision of your approximations can be adjusted for different applications, making these properties not only theoretical but also practical.