Problem 44 - Algebra 2

Question

In \(\Delta R S T, t=7 \mathrm{ft}\) and \(s=13 \mathrm{ft}\) . Find each value to the nearest tenth. Find \(r\) for \(m \angle R=35^{\circ}\)

Step-by-Step Solution

Answer

Based upon the problem, it's calculated that the length of side 'r' in triangle RST is approximately 9.6 ft when rounded to the nearest tenth.

Step 1: Identify the knowns and unknowns

In this problem, the length of side 't' is 7 ft, the length of side 's' is 13 ft, the measure of the angle R is 35 degrees, and we are to find the length of side 'r'.

Step 2: Use the sine rule

Given that the sides 's' and 't' are not directly opposite to the acute angle 35°, we cannot directly apply the sine rule. Instead, use the property that the sum of the angles in a triangle equals 180° to find the third angle. \n In \(\Delta R S T\), we know it is a right triangle because of Pythagorean triples (5,12,13). So, \(m \angle R = 35^\circ\), \(m \angle T = 90^\circ\), and \(m \angle S = 180^\circ - 90^\circ - 35^\circ = 55^\circ\). \n Then apply the sine rule which states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. \n By applying the sine rule, \(\frac{r}{sinR} = \frac{s}{sinS}\). In this case 'r' is the unknown we are trying to solve for. After rearranging, we have \(r = s \cdot \frac{sinR}{sinS}\).

Step 3: Calculate using the given values

Plugging in the given values \(r = 13 \cdot \frac{sin35^\circ}{sin55^\circ}\). Use a calculator to find the values of sin35 and sin55, then perform the required multiplication and division to get the value of 'r'.

Step 4: Round to the nearest tenth

After doing the arithmetic, we ensure to round the result to the nearest tenth as the instruction suggests.

Key Concepts

Triangle Angles

Understanding the angles in a triangle is a fundamental concept when working with triangles. Each triangle has three angles, and there is a key rule regarding their sum 🡪 all the angles in a triangle add up to 180 degrees. This principle is helpful when trying to find the measure of an unknown angle when the other angles are known.
In the given exercise, triangle \( \Delta R S T \) has \( m \angle R = 35^\circ \), and \( m \angle T = 90^\circ \). To find \( m \angle S \), we use the formula:
\[ m \angle S = 180^\circ - m \angle R - m \angle T \]
Plugging in the known values yields \( m \angle S = 55^\circ \). This step is crucial for later calculations when using other principles like the Law of Sines.

Law of Sines

The Law of Sines is a powerful tool in trigonometry for solving triangles. It relates triangle sides and their opposite angles. When one side and two angles of a triangle are known, this law can be used to find unknown sides.
It states that:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
In the context of the exercise, since we have two sides \( s = 13 \) ft and \( t = 7 \) ft, and angles \( m \angle R = 35^\circ \), \( m \angle S = 55^\circ \), the Law of Sines allows us to find the unknown side \( r \). By setting up the equation:
\[\frac{r}{\sin 35^\circ} = \frac{13}{\sin 55^\circ} \]
We are able to rearrange the formula for \( r \) and perform calculations to solve for the unknown side length.

Triangle Sides

The sides of a triangle are often what we're interested in finding, especially when given angles or other side lengths. Knowing the triangle types can be important in solving for side lengths. A right triangle, for example, which has one \( 90^\circ \) angle can make use of the Pythagorean theorem. However, the problem at hand uses trigonometric relationships due to known angles.
We used the Law of Sines, combining it with the known side and angle measurements. Given \( s = 13 \) ft and the opposite angles \( m \angle R = 35^\circ \) and \( m \angle S = 55^\circ \), we applied the formula to solve for side \( r \). The calculations involved using the sine function for the given angles and rearranging the equation to isolate \( r \).
Once you calculate the value of \( r = 13 \cdot \frac{\sin 35^\circ}{\sin 55^\circ} \), use a calculator to finalize the computation. Don't forget to round to the nearest tenth as instructed in the problem scenario!