Problem 15 - Algebra 2 and Trigonometry

Question

In \(\triangle C D E, \mathrm{m} \angle D=125, \mathrm{m} \angle E=28,\) and \(d=12.5 .\) Find \(c\) to the nearest hundredth.

Step-by-Step Solution

Answer

Side c is approximately 6.93 units.

Step 1: Determine the Measure of Angle C

In any triangle, the sum of all internal angles must be 180 degrees. We can use this property to find the measure of angle C in triangle CDE. Let m∠C be denoted as \( \mathrm{m} \angle C \). So we have: \( \mathrm{m} \angle C + \mathrm{m} \angle D + \mathrm{m} \angle E = 180 \). Substituting the given values \( 125^\circ \) for \( \mathrm{m} \angle D \) and \( 28^\circ \) for \( \mathrm{m} \angle E \), we get: \( \mathrm{m} \angle C = 180 - 125 - 28 = 27 \) degrees.

Step 2: Identify the Applicable Law or Formula

We are given side \( d \) opposite angle \( D \) and we need to find side \( c \), which is opposite angle \( C \). We can use the Law of Sines, which states: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). In this triangle: \( \frac{d}{\sin D} = \frac{c}{\sin C} \). We will use this relationship with the known measures of angles D and C, and the known length of side d to find c.

Step 3: Apply the Law of Sines

Using the Law of Sines: \( \frac{c}{\sin C} = \frac{d}{\sin D} \). We substitute the known values: \( \sin 27^\circ \) for \( \sin C \), \( \sin 125^\circ \) for \( \sin D \), and \( 12.5 \) for \( d \). This gives: \( \frac{c}{\sin 27} = \frac{12.5}{\sin 125} \).

Step 4: Solve for Side c

Reorganize the equation to solve for c: \( c = \frac{12.5 \times \sin 27}{\sin 125} \). Using a calculator, find the values: \( \sin 27 \approx 0.454 \) and \( \sin 125 \approx 0.819 \). Substitute these values into the equation: \( c = \frac{12.5 \times 0.454}{0.819} \approx 6.93 \).

Step 5: Conclusion

Round the value of c to the nearest hundredth: \( c \approx 6.93 \). Therefore, in triangle CDE, side c is approximately 6.93 units long.

Key Concepts

Triangle Angle Sum

One of the fundamental concepts of geometry is the Triangle Angle Sum. This principle states that the sum of the internal angles of a triangle is always 180 degrees. It's like a magic number for triangles. No matter how the triangle looks — whether it's tall and skinny or short and wide — the internal angles will always add up to 180 degrees.

Why does this matter? Knowing this principle is crucial. It helps you find unknown angles easily. If you have a triangle and you know two of the angles, you can easily calculate the third by subtracting the sum of the known angles from 180.

In our original exercise, we used this knowledge to find the missing angle, C, in triangle CDE. Given that angles D and E were 125 and 28 degrees respectively, we calculated angle C by subtracting the sum of those angles from 180:

• 180 - 125 - 28 = 27 degrees

This simple application of the Triangle Angle Sum is a handy tool when working with triangles in geometry.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It's particularly useful for solving triangles, meaning finding unknown side lengths or angles. One of the basic tools in trigonometry is the use of sine, cosine, and tangent functions.

In our exercise, trigonometry played a vital role. Specifically, we used the sine function from the Law of Sines. This law connects the angles and sides of a triangle and is particularly useful when you have some, but not all, information about a triangle.

For example, to find the length of side c of triangle CDE, we set up the relationship:

• \( \frac{d}{\sin D} = \frac{c}{\sin C} \)

Here, d is known, and D and C are angles provided or calculated in prior steps, allowing us to compute for c using trigonometric principles. By substituting the values of \( \sin 27 \) and \( \sin 125 \) into this equation, we found the length of side c.

Solving Triangles

Solving triangles involves determining unknown side lengths or angles based on the information you have. This can be done using several methods, including the Law of Sines and the Law of Cosines, both of which are part of trigonometry.

The goal is to make use of known measures to find unknowns, just like a puzzle. In our exercise, we solved for side c using the Law of Sines. Here's how it was done:

• We set up the equation \( \frac{c}{\sin 27} = \frac{12.5}{\sin 125} \).
• We rearranged the equation to solve for c: \( c = \frac{12.5 \times \sin 27}{\sin 125} \).
• We used a calculator to find the sine values to make the calculations easier.
• Finally, we found that c is approximately 6.93 when rounded to the nearest hundredth.

Understanding the process of solving triangles can be helpful in many real-world applications, such as engineering, architecture, and even daily problem-solving scenarios.