Question

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=42 \mathrm{~cm}, \mathrm{BC}=37 \mathrm{~cm}, \mathrm{AC}=26 \mathrm{~cm}\)

Short answer

Question: Given a triangle ABC with side lengths AB = 42, BC = 37, and AC = 26. Find the interior angles at vertices A, B, and C. Answer: To find the interior angles at vertices A, B, and C, follow these steps: 1. Calculate angle A using the cosine rule. 2. Calculate angle B using the sine rule. 3. Calculate angle C using the angle sum theorem. 4. State the angles A, B, and C as the final solution.

Step-by-step solution

Calculate angle A using the cosine rule

Using the cosine rule, we can find angle A knowing the sides AB, BC, and AC. The formula for angle A is: \[A = \arccos{\frac{B^2 + C^2 - A^2}{2BC}}\] Substitute the values: \[A = \arccos{\frac{(37)^2 + (26)^2 - (42)^2}{2(37)(26)}}\] Evaluate the arccos expression to find the angle A in degrees.

Calculate angle B using the sine rule

Having found angle A, we can use the sine rule to find angle B. The sine rule states that: \[\frac{a}{\sin{A}} = \frac{b}{\sin{B}}\] Rearrange to find \(\sin{B}\): \[\sin{B} = \frac{b \sin{A}}{a}\] Substitute the values: \[\sin{B} = \frac{(37) \sin{A}}{(42)}\] Evaluate the expression to find \(\sin{B}\) and use the arcsine function to find angle B in degrees.

Calculate angle C using the angle sum theorem

Since the sum of the angles in a triangle equals 180 degrees, we can find angle C by subtracting both angles A and B from 180 degrees: \[C = 180^\circ - A - B\] Substitute the values of angle A and angle B to find angle C in degrees.

State the solution

We have successfully found the angles A, B, and C in the triangle ABC. The final solution is the triangle with vertices A, B, and C, their respective side lengths, and their respective angle measures.

Key concepts

Cosine Rule

The Cosine Rule is a valuable tool in triangle calculations, enabling you to find the angles or sides of any triangle, not just right-angled ones. It's particularly useful when you know:

• All three sides of a triangle (SSS case), or
• Two sides and the included angle (SAS case).

The rule itself is an extension of the Pythagorean theorem and is expressed as:\[A = \arccos{\left( \frac{b^2 + c^2 - a^2}{2bc} \right)}\]
This formula calculates an angle by taking the sides of the triangle into account. The beauty of the cosine rule is its flexibility. It helps you "break the wall" of unknown angles when dealing with non-right triangles. Simply plug in your known side lengths, and you'll be able to solve for your desired angle. Apply this rule when your triangle's configuration leaves traditional methods fruitless.

Sine Rule

The Sine Rule is another key concept in solving triangles, especially handy when you have a mix of angles and sides. Unlike the cosine rule, which directly gives you an angle or a side, the sine rule works better with at least one complete ratio: either an angle and its opposite side, or vice versa. The formula is:\[\frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}}\]This equation shows the relationship between the lengths of the sides of a triangle and the sines of its angles.
Using the sine rule is intuitive once you have at least part of one ratio. Rearrange the equation to solve for the unknown by isolating the desired term. It's particularly effective when:

• Two angles and a side are known (AAS or ASA).
• An angle and the corresponding side, along with another side, are known (SSA).

Remember, the sine rule is all about matching angles and opposite sides.

Angle Sum Theorem

The Angle Sum Theorem is a fundamental principle of geometry, stating that the sum of the interior angles of any triangle is always 180 degrees. This theorem forms the foundation of many triangle-related problems and solutions.Whether working with a right-angled, isosceles, or scalene triangle, the angle sum theorem holds true. It provides a straightforward method to find the last unknown angle when two angles are already known. Simply subtract the sum of the two known angles from 180 degrees:\[C = 180^\circ - A - B\]
Where \(A\) and \(B\) are the known angles. This method makes solving for the third angle quick and reliable.
Utilizing this theorem allows for rapid verification of your calculated angles, ensuring they align with the inherent properties of triangles. It’s a brilliant example of how understanding basic principles leads to efficient problem-solving in geometry.