Problem 41 - Algebra 2

Question

REVIEW The longest side of a triangle is 67 inches. Two angles have measures of \(47^{\circ}\) and \(55^{\circ} .\) What is the length of the shortest leg of the triangle? $$ \begin{array}{ll}{\text { F } 50.1 \text { in. }} & {\text { H } 60.1 \text { in. }} \\ {\text { G } 56.1 \text { in. }} & {\text { J } 62.3 \text { in. }}\end{array} $$

Step-by-Step Solution

Answer

The shortest leg is 50.1 inches (Option F).

Step 1: Calculate the Third Angle

Use the triangle angle sum property, which states that the sum of angles in a triangle is always \(180^{\circ}\). Given two angles \(47^{\circ}\) and \(55^{\circ}\), the third angle \(C\) can be calculated as follows: \[ C = 180^{\circ} - 47^{\circ} - 55^{\circ} = 78^{\circ}. \]

Step 2: Identify the Longest Side as the Hypotenuse

The given side, 67 inches, is the longest side and therefore, the hypotenuse of the triangle if it were a right-angled triangle, but since none of the angles is 90 degrees, it can't be a right triangle. However, given the largest angle is \(78^{\circ}\), the opposite side will be less than 67 inches.

Step 3: Apply the Law of Sines

Utilize the Law of Sines, which states: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}. \] Let side \(a\) opposite to angle \(A = 47^{\circ}\), side \(b\) opposite to angle \(B = 55^{\circ}\), and side \(c = 67\) inches opposite angle \(C = 78^{\circ}\). Solve for the shortest leg by checking side \( a \), opposite the smallest angle.

Step 4: Calculate the Shortest Leg

Rearrange the Law of Sines to solve for \(a\): \[ a = c \cdot \frac{\sin A}{\sin C} = 67 \cdot \frac{\sin 47^{\circ}}{\sin 78^{\circ}}. \] Compute \(a\) using sine values, \( \sin 47^{\circ} \approx 0.7314\) and \( \sin 78^{\circ} \approx 0.9781\): \[ a \approx 67 \cdot \frac{0.7314}{0.9781} \approx 50.1 \text{ inches}. \]

Step 5: Verify Result with Answer Options

Now check the computed length against the choices provided: \(F\) 50.1 in., \(H\) 60.1 in., \(G\) 56.1 in., \(J\) 62.3 in. The correct answer matches choice \(F\).

Key Concepts

Triangle Properties

A triangle is a three-sided polygon characterized by three angles and three sides. Triangles can vary in shape and size but always adhere to some core properties. These properties ensure that no matter how irregular a triangle may look, it has a structure governed by mathematical principles.

• Every triangle has three sides and three angles.
• The sum of its interior angles is always constant at 180 degrees.
• Types of triangles include equilateral, isosceles, and scalene, defined by their sides.

Understanding these properties helps in identifying how different parts of a triangle, including angles and sides, interact with each other. This is especially useful when using geometric methods like the Law of Sines, which helps in determining unknown lengths or angles.

Angle Sum Property

The Angle Sum Property is a fundamental aspect of triangles. It states that the sum of the interior angles of any triangle is exactly 180 degrees, making it a crucial element in solving many geometry problems.
In the problem provided, we have two angles measuring 47° and 55°. To find the third angle, we simply subtract the sum of the given angles from 180°:
\[ A + B + C = 180° \]
where \( A = 47° \), \( B = 55° \), so \( C = 180° - 47° - 55° = 78° \).
This property not only helps us find the missing angle but also reassures us that our triangle fits the basic definition, having interior angles sum to 180 degrees.

Sine Function

The sine function is a trigonometric function that relates the angle of a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. It is often abbreviated as sin.
The sine function is essential in understanding the proportional relationships between the sides and angles of a triangle.

• For any angle θ in a right triangle, \( ext{sin}( heta) = \) \(\frac{\text{Opposite}}{\text{Hypotenuse}} \).
• The sine function varies between -1 and 1, and it is maximized at angles of 90° or \( \frac{\pi}{2} \, ext{radians} \).

In the context of the Law of Sines, sine values for the angles of non-right triangles help determine the proportional lengths of sides. In our solution, knowing the sine values for the angles allows us to compute the shortest side of a triangle accurately.

Triangle Sides

The sides of a triangle play a vital role in determining its overall geometry and properties. The Law of Sines provides an effective means to analyze these sides based on the triangle's angles.
In general, the relationship between the sides and angles can be expressed mathematically, such as in the Law of Sines formula:
\[ \frac{a}{ ext{sin}A} = \frac{b}{ ext{sin}B} = \frac{c}{ ext{sin}C} \]
Where \( a, b, \) and \( c \) are the sides opposite angles \( A, B, \) and \( C \) respectively.
In the given exercise, the task was to find the shortest leg opposite the angle of 47° using the Law of Sines. With side \( c = 67 \) inches and the calculated \ \, it simplifies finding the unknown side. The approach ensures precision and accuracy, reinforcing how the sides' lengths can be determined from the given angles and one known side. This closes the gap between understanding abstract geometric concepts and practical problem-solving.