Problem 38 - Intermediate Algebra

Question

Find the missing lengths in each triangle. Give the exact answer and then an approximation to two decimal places. See Example 5. 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 \( 8\sqrt{3} \) (≈ 13.86 yards) and the hypotenuse is \( 16\sqrt{3} \) (≈ 27.71 yards).

Step 1: Understanding the Triangle Ratios

In a 30°-60°-90° triangle, the sides opposite these angles have a special ratio. The side opposite the 30° angle (shorter leg) is half the hypotenuse, the side opposite the 60° angle (longer leg) is \( \sqrt{3} \) times the shorter leg, and the hypotenuse is twice the shorter leg.

Step 2: Define Variables

Let the shorter leg be \( x \). According to the ratio properties, the longer leg, which is opposite the 60° angle, is \( x \sqrt{3} \). We are given that the longer leg is 24 yards, so \( x \sqrt{3} = 24 \).

Step 3: Solve for the Shorter Leg

Solve for \( x \) in the equation \( x \sqrt{3} = 24 \). Divide both sides by \( \sqrt{3} \):\[ x = \frac{24}{\sqrt{3}} \]Multiply numerator and denominator by \( \sqrt{3} \) to rationalize the denominator:\[ x = \frac{24}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{24\sqrt{3}}{3} = 8\sqrt{3} \]

Step 4: Calculate the Hypotenuse

The hypotenuse is twice the length of the shorter leg \( x \). Using \( x = 8\sqrt{3} \), the hypotenuse \( h \) is:\[ h = 2 \times 8\sqrt{3} = 16\sqrt{3} \]

Step 5: Approximate the Results

Now, approximate the values using the known value of \( \sqrt{3} \approx 1.732 \). Calculate:- Shorter leg: \[ 8\sqrt{3} \approx 8 \times 1.732 = 13.856 \approx 13.86 \] - Hypotenuse: \[ 16\sqrt{3} \approx 16 \times 1.732 = 27.712 \approx 27.71 \]

Step 6: Conclusion

The exact lengths are the shorter leg \( 8\sqrt{3} \) and the hypotenuse \( 16\sqrt{3} \). The approximations to two decimal places are 13.86 yards and 27.71 yards, respectively.

Key Concepts

Understanding Triangle Ratios in a 30°-60°-90° Triangle

A 30°-60°-90° triangle is special because the sides are always in a set ratio. This specific kind of triangle always has its sides in the following order:
- The side opposite the 30° angle is the shortest. This side is known as the shorter leg.- The hypotenuse, which is the side opposite the right angle, is the longest side.- The side opposite the 60° angle is the longer leg.

In these triangles, this ratio can be memorized with simple relationships:

• The shorter leg is 1/2 of the hypotenuse.
• The longer leg equals the shorter leg times \( \sqrt{3} \).
• The hypotenuse is twice as long as the shorter leg.

If you're given the length of one side, you can use these ratios to find the other two. Learning these ratios can save you time and effort when dealing with these types of right triangles.

Solving Right Triangles with Given Sides

To solve a right triangle, particularly a 30°-60°-90° triangle, you start by identifying what information you have and what you need to find.

In this exercise:

• We are given the longer leg's length, 24 yards, which is opposite the 60° angle.
• We need to find both the hypotenuse and the shorter leg.

First, we define variables to make our work easier. Let's denote the shorter leg as \( x \). According to the triangle's ratios, we know:

• Longer leg = \( x \sqrt{3} \)
• Given that \( x \sqrt{3} = 24 \)

We solve for \( x \) by isolating it. This means dividing both sides by \( \sqrt{3} \):

\[ x = \frac{24}{\sqrt{3}} \]

To rationalize the denominator, multiply both numerator and denominator by \( \sqrt{3} \):

\[ x = \frac{24\sqrt{3}}{3} = 8\sqrt{3} \]

Then, use the value of \( x \) to find the hypotenuse:

Hypotenuse = \( 2x = 16\sqrt{3} \)

This solution method has now found the lengths you needed by leveraging ratio knowledge.

Understanding Exact Answers and Approximations

While exact answers provide precision using radicals like \( \sqrt{3} \), sometimes we need decimal approximations. This helps when you need to understand how these lengths might look or feel in real-world measurements.

To approximate, you use a known decimal value for \( \sqrt{3} \), commonly about 1.732. Then follow these steps:

• For the shorter leg:

  \[ 8\sqrt{3} \approx 8 \times 1.732 = 13.856 \approx 13.86 \]

• For the hypotenuse:

  \[ 16\sqrt{3} \approx 16 \times 1.732 = 27.712 \approx 27.71 \]

Using both exact and approximate values allows you to better discuss and visualize scenarios. Keep in mind that when reporting results or solving real-life problems, knowing both can be very helpful.