Question

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=29 \mathrm{~cm}, \mathrm{BC}=41 \mathrm{~cm}, B=100^{\circ}\)

Short answer

Question: In triangle ABC, given side AB = 29 cm, side BC = 41 cm, and angle B = 100°, find the remaining components: side AC, angle A, and angle C. Solution: By applying the Law of Cosines and Law of Sines, we found the following missing components of triangle ABC: - Side AC ≈ 51.15 cm - Angle A ≈ 42.49° - Angle C ≈ 37.51°

Step-by-step solution

Identify the given information

In this problem, we are given: - Side AB = 29 cm - Side BC = 41 cm - Angle B = 100° We have the task to find the remaining components of the triangle: - Angle A - Angle C - Side AC

Use the Law of Cosines to find side AC

First, we will use the Law of Cosines to find the third side length, AC. Using the given data, the Law of Cosines formula becomes: \[AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(B)\] Substitute the values given: \[AC^2 = 29^2 + 41^2 - 2 \cdot 29 \cdot 41 \cdot \cos(100^{\circ})\] Now, we can calculate the value of AC: \[AC \approx 51.15\] So side AC is approximately 51.15 cm.

Use the Law of Sines to find angle A

To find angle A, we can use the Law of Sines: \[\frac{\sin{A}}{AB} = \frac{\sin{B}}{BC}\] Rearrange the formula to solve for angle A: \[\sin{A} = \frac{AB \cdot \sin{B}}{BC}\] Now, substitute the given values: \[\sin{A} = \frac{29 \cdot \sin{100^{\circ}}}{41}\] Calculate the value of sin(A) and then find the measure of angle A: \[A \approx 42.49^{\circ}\] Angle A is approximately 42.49°.

Use the angle sum property to find angle C

Now we can find angle C using the angle sum property of triangles, which states that the sum of the three angles in a triangle is equal to 180°. Therefore: \[A + B + C = 180^{\circ}\] Substitute the values of angle A and angle B: \[42.49^{\circ} + 100^{\circ} + C = 180^{\circ}\] Now, solve for angle C: \[C \approx 37.51^{\circ}\] So angle C is approximately 37.51°.

Present the final solution

Now we have found all the missing components of the triangle: - Side AC ≈ 51.15 cm - Angle A ≈ 42.49° - Angle C ≈ 37.51° Thus, we have successfully solved triangle ABC.

Key concepts

Law of Sines

The Law of Sines is a powerful tool for solving triangles, particularly when you know an angle and its opposite side, or when you need to find a missing angle. It relates the sides and angles of a triangle in a very elegant way. The formula looks like this:

• \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\)

Here, \(A\), \(B\), and \(C\) are the angles of the triangle, whereas \(a\), \(b\), and \(c\) are the lengths of the sides opposite to these angles.
In our exercise, we used the Law of Sines to determine angle A. Given side AB, angle B, and side BC, we could write the expression \(\frac{\sin A}{29} = \frac{\sin 100^\circ}{41}\) to find \(A\). This formula is incredibly useful whenever you're missing an angle or need to find a side length and have at least one complete ratio to work with.

Triangle Solving

Solving a triangle means finding the unknown side lengths and angles when certain elements are given. In the context of our exercise, we started with a side-angle-side (SAS) scenario – having two sides and the included angle. This called for the use of the Law of Cosines initially to find the third side length.
Once we found side AC using the Law of Cosines, our next aim was to find the remaining unknown angles using the Law of Sines and the angle sum property.

• From knowing side AB, side BC, and angle B, we calculated angle A using the sines' law.
• This sequential approach of utilizing different mathematical tools efficiently helped us in deriving the full scope of the triangle's dimensions and properties.

Triangle solving often requires a combination of methods to reach the solution, which demonstrates the interconnected nature of geometric principles.

Angle Sum Property

The angle sum property is a fundamental principle of triangle geometry that is straightforward yet indispensable. It states that the sum of the internal angles of any triangle always equals 180 degrees. This property serves as a final verification step in solving triangles.
In our example, after finding angle A using the Law of Sines, we applied this rule to determine angle C. We already knew angle B as 100 degrees and calculated angle A to be approximately 42.49 degrees. Thus,

• The equation became \(42.49^\circ + 100^\circ + C = 180^\circ\).

Solving this allows us to find the unknown angle C, which ties up any loose ends from previous calculations. The angle sum property not only helps in solving for missing angles but also acts as a check to ensure that all calculated angles are consistent with the inherent properties of triangles.