Problem 41 - Elementary and Intermediate Algebra

Question

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) right triangle, the length of the hypotenuse is 1.5 feet. To the nearest hundredth, find the length of the shorter leg and the length of the longer leg. Give the exact answer and then an approximation to two decimal places, when appropriate.

Step-by-Step Solution

Answer

Shorter leg: 0.75 feet, Longer leg: 1.30 feet

Step 1: Understand the Triangle Properties

A \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle has sides in the ratio \(1:\sqrt{3}:2\). The shortest side (opposite \(30^{\circ}\)) is \(\frac{1}{2}\) of the hypotenuse. The longer side (opposite \(60^{\circ}\)) is \(\frac{\sqrt{3}}{2}\) of the hypotenuse.

Step 2: Calculate Shorter Leg

The shorter leg, opposite the \(30^{\circ}\) angle, is found using the formula: \(\text{Shorter Leg} = \frac{1}{2} \times \text{Hypotenuse}\). In this case, it is \(\frac{1}{2} \times 1.5 = 0.75\) feet, exactly.

Step 3: Calculate Longer Leg

The longer leg, opposite the \(60^{\circ}\) angle, is found using the formula: \(\text{Longer Leg} = \frac{\sqrt{3}}{2} \times \text{Hypotenuse}\). In this case, it is \(\frac{\sqrt{3}}{2} \times 1.5 \, \text{feet}\). This simplifies to \(\frac{3\sqrt{3}}{4}\) feet, exactly.

Step 4: Approximations

Using a calculator, \(\sqrt{3} \approx 1.732\). Thus, the longer leg can be approximated as \(\frac{3 \times 1.732}{4} \approx 1.30\) feet, rounded to two decimal places.

Key Concepts

Understanding the Hypotenuse

In a right triangle, the hypotenuse is the longest side and it's always opposite the right angle. For a 30-60-90 triangle, this means knowing the hypotenuse helps to easily find the other two sides using specific ratios. In such a triangle, the hypotenuse is twice as long as the shortest leg (opposite the 30° angle).

So, if we have the hypotenuse as 1.5 feet, then we can directly use this relationship to determine the exact lengths of the other sides.

• The shortest leg is half of the hypotenuse, so it's 0.75 feet.
• This relationship and pattern keeps calculations simple.

Understanding the hypotenuse in the 30-60-90 triangle leads to quick and exact results right from the start!

Exploring Trigonometric Ratios

Trigonometric ratios are essential in triangles, especially in right triangles like the 30-60-90 triangle, as they allow for a clear relationship between each angle and its opposite sides.

Key ratios in the 30-60-90 triangle are:

• The sine of 30° is 0.5, which corresponds to the ratio of the shorter leg over the hypotenuse.
• The cosine of 60° is also 0.5, supporting the symmetry of our triangle.
• The tangent of 30° is \(\frac{1}{\sqrt{3}}\), associating the opposite side to the adjacent side.

These ratios, when known, allow seamless transitions from angles to side lengths, providing a robust method to determine triangle dimensions. They simplify complex problems into easily manageable calculations!

Exact Values and Approximations

Exact values and approximations are pivotal when solving geometric problems. In the case of the 30-60-90 triangle, both play distinct roles.

Exact values are grounded in the unique proportional relationships:

• For example, the longer leg is exactly \(\frac{\sqrt{3}}{2}\) times the hypotenuse.
• In our case, this equates to \(\frac{3\sqrt{3}}{4}\) feet exactly.

Approximations become important when exact radical values are cumbersome for everyday use or context:

• Since \(\sqrt{3}\) is about 1.732, the longer leg can be approximated to 1.30 feet.
• These approximations are handy, providing clarity while still being neatly rounded to two decimal places.

Thus, a mix of these values lets you maintain precision or ensure practical usability when needed.