Problem 38 - Algebra 2

Question

In \(\triangle A B C, m \angle A=40^{\circ}\) and \(m \angle B=30^{\circ} .\) Find each value to the nearest tenth. Find \(A C\) for \(B C=10.5 \mathrm{m}\)

Step-by-Step Solution

Answer

AC measures approximately 13.7 meters.

Step 1: Understanding the Given Information

We are given a triangle ABC with angles A and B measuring 40 degrees and 30 degrees, respectively. Using the fact that the sum of angles in a triangle is 180 degrees, we can find angle C.

Step 2: Finding the Measure of Angle C

Subtract the sum of the measures of angles A and B from 180 degrees to find the measure of angle C: \(m \angle C = 180^\circ - m \angle A - m \angle B = 180^\circ - 40^\circ - 30^\circ = 110^\circ\).

Step 3: Applying the Law of Sines

To find side AC, we can apply the Law of Sines which states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in any triangle. \(\frac{AC}{\sin(B)} = \frac{BC}{\sin(A)}\)

Step 4: Calculate the Length of AC

Rearrange the formula and solve for AC: \(AC = \frac{BC \cdot \sin(A)}{\sin(B)} = \frac{10.5 \cdot \sin(40^\circ)}{\sin(30^\circ)}\). Use a calculator to find the sine values and compute AC.

Step 5: Finding AC to the Nearest Tenth

After calculation, you'll get AC's length and then round it to the nearest tenth. Please ensure that your calculator is set to the correct mode (degrees) to get an accurate result.

Key Concepts

Triangle angle sum theorem

Understanding the Triangle Angle Sum Theorem is essential when working with any type of triangle problems. This theorem states that the sum of the interior angles of any triangle will always add up to 180 degrees. No matter how you shape your triangle, the three angles inside it must conform to this rule. For example, if two angles are known, like in our exercise where \(m \angle A = 40^\circ\) and \(m \angle B = 30^\circ\), we can always find the third angle by subtracting the sum of the known angles from 180 degrees.

Let's apply this to our problem: We have \(m \angle C = 180^\circ - m \angle A - m \angle B = 180^\circ - 40^\circ - 30^\circ = 110^\circ\). The ability to find a missing angle is not just useful for triangle problems but is a foundation for understanding more advanced geometry concepts.

Solving triangles

Solving triangles involves finding the missing sides and angles in a triangle when certain sides and angles are given. This process uses various methods, including the Triangle Angle Sum Theorem, the Pythagorean Theorem for right-angled triangles, and trigonometric ratios for non-right triangles. The Law of Sines and Law of Cosines are particularly important for solving triangles that do not contain a right angle.

The exercise we're examining gave us two angles and one side. Our goal was to find another side of the triangle, which is a classic example of solving a triangle. By applying the correct principles, you can calculate all unknown aspects of a triangle, as we did with the angle and will do with the side \( AC \).

Trigonometric ratios

Trigonometric ratios provide a relationship between the angles and sides of a triangle. For a right triangle, the fundamental ratios are sine (sin), cosine (cos), and tangent (tan). These ratios associate the lengths of a triangle's sides with its angles. For example, the sine of an angle in a right triangle is the ratio of the length of the opposite side to that angle divided by the length of the triangle's hypotenuse.

Using Trigonometric Ratios

In the context of our exercise, we use the sine function to compare the ratios of two sides of a triangle to their opposite angles. Using this ratio allows us to solve for an unknown side when given an angle and another side. Trigonometric ratios are not just tools for solving triangles; they are crucial in fields such as physics, engineering, and in various applications involving waves and oscillations.

Sine rule application

The Sine Rule, also known as the Law of Sines, is a powerful principle that applies to any triangle, whether it's a right-angled triangle or not. The rule states that the ratio of the length of a side to the sine of its opposite angle is the same for all sides and angles in a triangle. It can be written as \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \), where \(a, b,\) and \(c\) are the sides of the triangle, and \(A, B,\) and \(C\) are the respective opposite angles.

Finding Missing Sides

In our example, we find side \( AC \) by isolating it in the equation: \( AC = \frac{BC \cdot \sin(A)}{\sin(B)} = \frac{10.5 \cdot \sin(40^\circ)}{\sin(30^\circ)} \). After calculating the sine values and plugging them into the formula, we can easily determine the length of \( AC \). The utility of the Sine Rule extends beyond geometry, aiding in various real-world problems that require precise measurements, such as in navigation and surveying.