Problem 13 - Algebra 2 and Trigonometry

Question

In \(\triangle P Q R, \mathrm{m} \angle P=125, \mathrm{m} \angle Q=14,\) and \(p=122 .\) Find \(r\) to the nearest integer.

Step-by-Step Solution

Answer

The side \( r \) is approximately 98.

Step 1: Understanding the Angles in a Triangle

First, recall the fact that the sum of the angles in any triangle is always equal to 180 degrees. This can be expressed as \( \angle P + \angle Q + \angle R = 180 \). We are given \( \angle P = 125 \) and \( \angle Q = 14 \).

Step 2: Finding the Third Angle

Use the information from Step 1 to find \( \angle R \) by calculating \( \angle R = 180 - 125 - 14 \). This simplifies to \( \angle R = 41 \).

Step 3: Using the Sine Rule

Now that we have all the angles, apply the sine rule which is \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We substitute \( a = 122 \), \( A = 125^\circ \), \( C = 41^\circ \) to find \( r \), the side opposite \( \angle R \).

Step 4: Solving for \( r \) Using Sine Rule

From Step 3, the equation with the sine rule becomes \( \frac{122}{\sin 125^\circ} = \frac{r}{\sin 41^\circ} \). Solve for \( r \) by cross-multiplying which gives \( r = \frac{122 \sin 41^\circ}{\sin 125^\circ} \).

Step 5: Calculating \( r \)

Use a calculator to determine the sine values: \( \sin 125^\circ \approx 0.819 \) and \( \sin 41^\circ \approx 0.656 \). Therefore, \( r \approx \frac{122 \times 0.656}{0.819} \approx 97.7 \).

Step 6: Rounding to the Nearest Integer

Finally, round \( 97.7 \) to the nearest integer to find that \( r = 98 \).

Key Concepts

Triangle Angles

When dealing with triangles, one of the most fundamental concepts is understanding the properties of their angles. A triangle, being a three-sided polygon, always has three internal angles. A key rule that applies to all triangles, regardless of type, is that the sum of these angles is always 180 degrees. This is known as the Triangle Angle Sum Property. So, if you know two angles of a triangle, you can always calculate the third angle using the formula: \( \angle P + \angle Q + \angle R = 180 \).
Knowing how to determine an unknown angle when the other two are given is a basic yet crucial skill for solving many triangle-related problems. For example, in our exercise, we calculated \( \angle R \) by subtracting the sum of \( \angle P \) and \( \angle Q \) from 180. This principle not only applies to problems on paper but is also useful in practical scenarios, like architectural designs and engineering projects.
Understanding this concept simplifies many problems and reduces complex questions to simpler forms, making it easier to handle more intricate mathematical challenges involving triangles.

Angle Sum

The concept of angle sum, specifically within the context of triangles, forms the foundational basis for much of trigonometry and geometry. This principle dictates that the angles within a triangle always add up to exactly 180 degrees. Understanding this allows one to determine an unknown angle if the other two are known. This is often the first step in solving many triangle-related problems. Here’s how it works:
- If you know two angles, simply subtract their sum from 180 to get the third angle.
- This rule holds true for all triangles, whether they are scalene, isosceles, or equilateral.

In practical exercises, like the one we're discussing, knowing this property is instrumental in finding missing measurements, which can then be used in further calculations, such as those involving the sine rule.
By internalizing the angle sum property, you lay a solid foundation for advanced geometric reasoning, enabling you to tackle more complex problems with confidence.

Trigonometry

Trigonometry, the branch of mathematics that deals with the relationships between the sides and angles of triangles, plays a vital role in solving triangle problems. One of the key tools in trigonometry is the Sine Rule, a formula used to find unknown sides or angles in any triangle, not just right-angled ones. The rule is given by \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \), where \( a, b, \) and \( c \) are the sides of the triangle, and \( A, B, \) and \( C \) are the opposite angles.
In our exercise, the Sine Rule was crucial for calculating the length of side \( r \) opposite \( \angle R \). Given \( p \) and the angles, we applied the formula and solved for \( r \), demonstrating how trigonometry enables us to find unknown dimensions using known parameters. This formula is particularly useful in non-right triangles where traditional trigonometric ratios like sine, cosine, and tangent of angles \( A \) and \( B \) might not apply directly.

• Using the Sine Rule: Substitute known values into the formula to find the unknown side or angle.
• Cross-multiply to solve for the unknown.
• Calculate the sine of given angles accurately using a calculator.
• Solve the equation to find the missing measurement.

Understanding and applying the Sine Rule in trigonometry is an invaluable skill for solving various geometric problems efficiently and accurately.