Question

For questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(A=36^{\circ}, B=79^{\circ}, \mathrm{AC}=11.63 \mathrm{~cm}\)

Short answer

Answer: The lengths of sides AB and BC are approximately \(6.96 \: \mathrm{cm}\) and \(10.32 \: \mathrm{cm}\), respectively.

Step-by-step solution

Find angle C

To find angle C, subtract the given angles A and B from \(180^{\circ}\): \(C = 180^{\circ} - 36^{\circ} - 79^{\circ} = 65^{\circ}\)

Use Law of Sines to find side AB

Using the Law of Sines, we can find the length of side AB: \(\frac{\mathrm{AB}}{\sin{A}} = \frac{\mathrm{AC}}{\sin{B}}\) $\mathrm{AB} = \frac{\mathrm{AC} \times \sin{A}}{\sin{B}} \Rightarrow \mathrm{AB} = \frac{11.63 \times \sin{36^{\circ}}}{\sin{79^{\circ}}} \Rightarrow \mathrm{AB} \approx 6.96 \: \mathrm{cm}$

Use Law of Sines to find side BC

Using the Law of Sines, we can find the length of side BC: \(\frac{\mathrm{BC}}{\sin{C}} = \frac{\mathrm{AC}}{\sin{B}}\) $\mathrm{BC} = \frac{\mathrm{AC} \times \sin{C}}{\sin{B}} \Rightarrow \mathrm{BC} = \frac{11.63 \times \sin{65^{\circ}}}{\sin{79^{\circ}}} \Rightarrow \mathrm{BC} \approx 10.32\: \mathrm{cm}$ Now we have found all the unknown sides and angles of the triangle. The triangle ABC can be described as: - Angle A: \(36^{\circ}\) - Angle B: \(79^{\circ}\) - Angle C: \(65^{\circ}\) - Side AC: \(11.63 \: \mathrm{cm}\) - Side AB: \(\approx 6.96 \: \mathrm{cm}\) - Side BC: \(\approx 10.32 \: \mathrm{cm}\)

Key concepts

Triangle Solving

Solving a triangle means finding all the missing sides and angles of the triangle when some are already known. In the context of this exercise, solving \( \triangle \mathrm{ABC} \) involves applying geometry principles such as the Law of Sines and the rule that the sum of angles in a triangle is always \(180^{\circ}\).

When given two angles and one side, known as the AAS (Angle-Angle-Side) situation, the problem is often solved by calculating the third angle first. This is because knowing all three angles allows for the effective application of the Law of Sines to find the unknown sides of the triangle.

Simplifying the process involves the following steps:

• Identify the known quantities: In the given problem, angles \(A\) \(=36^{\circ}\) and \(B = 79^{\circ}\) are known along with side \(AC = 11.63 \, \mathrm{cm}\).
• Calculate the missing angle \(C\) using the sum of angles in a triangle.
• Utilize the Law of Sines to find the remaining sides \(AB\) and \(BC\).

By following these steps, the entire triangle is solved quickly and accurately. This forms the foundation for further geometry problem-solving tasks.

Angle Calculation

Angle calculation in triangles revolves around one fundamental fact: the internal angles of a triangle always sum to \(180^{\circ}\). This principle can be applied in any triangle to find a missing angle given the other two.

In our exercise, we were tasked with finding angle \(C\). With angles \(A = 36^{\circ}\) and \(B = 79^{\circ}\) already known, calculating angle \(C\) is straightforward.

• Subtract the sum of angles \(A\) and \(B\) from \(180^{\circ}\).
• Thus, \(C = 180^{\circ} - 36^{\circ} - 79^{\circ} = 65^{\circ}\).

This technique is invaluable as it doesn't require any special formulas or tools, just basic arithmetic and an understanding of triangle properties. By always confirming the angles add up to \(180^{\circ}\), you can check your solution for potential errors.

Side Calculation

Calculating the sides of a triangle can become straightforward with the Law of Sines, especially if at least two angles and a side are known. This law relates the lengths of sides to the sines of their respective opposite angles, forming an important basis for side calculation in triangles.

For triangle \( \mathrm{ABC} \) in this exercise, the Law of Sines is applied as follows:

• First, calculate side \(AB\) using: \( \frac{\mathrm{AB}}{\sin{A}} = \frac{\mathrm{AC}}{\sin{B}} \). Solving this equation gives: \( \mathrm{AB} = \frac{11.63 \times \sin{36^{\circ}}}{\sin{79^{\circ}}} \approx 6.96 \, \mathrm{cm} \).
• Then, calculate side \(BC\) using: \( \frac{\mathrm{BC}}{\sin{C}} = \frac{\mathrm{AC}}{\sin{B}} \). This results in: \( \mathrm{BC} = \frac{11.63 \times \sin{65^{\circ}}}{\sin{79^{\circ}}} \approx 10.32 \, \mathrm{cm} \).

Accurate side calculation requires precise calculation of the sine values and correct use of the formula. Remember, rounding errors can affect the final value, so it's crucial to maintain accuracy throughout the calculations.