Problem 16 - Algebra 2 and Trigonometry

Question

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle P Q R, p=12, \mathrm{m} \angle Q=80,\) and \(\mathrm{m} \angle R=30\)

Step-by-Step Solution

Answer

Angle P is 70°, side q is approximately 13, and side r is approximately 6.

Step 1: Identify Known and Unknown Elements

We know side \( p = 12 \), angle \( \angle Q = 80^\circ \), and angle \( \angle R = 30^\circ \). We need to find angle \( \angle P \), and the sides \( q \) and \( r \).

Step 2: Find Angle P

Using the fact that the sum of the angles in a triangle is \( 180^\circ \), we calculate \( \angle P \) as follows: \[ \angle P = 180^\circ - \angle Q - \angle R = 180^\circ - 80^\circ - 30^\circ = 70^\circ \].

Step 3: Apply Sine Law for Side q

Using the sine law: \[ \frac{q}{\sin Q} = \frac{p}{\sin P} \] gives \[ q = \frac{12 \times \sin 80^\circ}{\sin 70^\circ}. \] Calculate \( q \) using \( \sin(80^\circ) \approx 0.9848 \) and \( \sin(70^\circ) \approx 0.9397 \), thus \( q \approx \frac{12 \times 0.9848}{0.9397} \approx 12.58 \), and round to the nearest integer: \( q \approx 13 \).

Step 4: Apply Sine Law for Side r

Similarly, applying the sine law for side \( r \): \[ \frac{r}{\sin R} = \frac{p}{\sin P} \] gives \[ r = \frac{12 \times \sin 30^\circ}{\sin 70^\circ}, \] where \( \sin(30^\circ) = 0.5 \). Calculate \( r \) as \( r \approx \frac{12 \times 0.5}{0.9397} \approx 6.39 \), and round to the nearest integer: \( r \approx 6 \).

Key Concepts

Triangle Solving

Solving a triangle means to find the missing parts of it, given certain known values. In our exercise, we have a triangle \(PQR\) with known side \(p\) and angles \(\angle Q\) and \(\angle R\). To solve this triangle, we must find the missing angle \(\angle P\) and the two sides \(q\) and \(r\). It's like playing detective with numbers and rules. We use given information and mathematical principles to uncover the unknown pieces of the puzzle. This process involves understanding and implementing key concepts such as the angle sum property and the sine law.

Sine Law

The Sine Law is a powerful tool that helps relate the sides and angles in a triangle. It's particularly useful when we have a combination of known angles and sides, like in our exercise. The Sine Law is given by the equation: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] This equation tells us that the ratio between the length of a side and the sine of its opposite angle is the same for each side of the triangle.

• For side \(q\), we use: \( \frac{q}{\sin Q} = \frac{p}{\sin P} \)
• For side \(r\), we use: \( \frac{r}{\sin R} = \frac{p}{\sin P} \)

Knowing the Sine Law allows us to accurately calculate the missing sides once we know all angles, making it a reliable method in triangle solving.

Angle Sum Property

The Angle Sum Property is a fundamental rule for triangles that states the sum of a triangle's interior angles always equals \( 180^\circ \). This property is essential when we have two angles and need to find the third, as in our exercise for \(\angle P\). Here’s how it works in practice:

• Start with the equation: \( \angle P + \angle Q + \angle R = 180^\circ \)
• Knowing \(\angle Q = 80^\circ\) and \(\angle R = 30^\circ\), substitute these values into the equation.
• Solve for \(\angle P\): \[ \angle P = 180^\circ - 80^\circ - 30^\circ = 70^\circ \]

This simple yet powerful property ensures you can always find a triangle's missing angle when two are known.

Triangle Sides Calculation

Calculating the sides of a triangle using known elements involves applying methods like the Sine Law when dealing with non-right triangles. In the given problem, once we determine all angles, we use the Sine Law to find the unknown sides.

• For side \(q\), calculate using: \[ q = \frac{12 \times \sin 80^\circ}{\sin 70^\circ} \] We approximate \(\sin 80^\circ \approx 0.9848\) and \(\sin 70^\circ \approx 0.9397\), leading to \(q \approx 13\) after rounding.
• For side \(r\), calculate using: \[ r = \frac{12 \times \sin 30^\circ}{\sin 70^\circ} \] With \(\sin 30^\circ = 0.5\), we find \(r \approx 6\) after rounding.

These steps allow us to complete the triangle, ensuring all sides and angles are consistent with the given and calculated values.