10.4. Detailed Worked Examples

1. Applying Law of Cosines to find unknown side

In this example, we apply the Law of Cosines, which helps us determine the length of an unknown side in a triangle when we know the lengths of the other two sides and the angle between them. The Law of Cosines states that for any triangle, if a, b, and c are the lengths of the sides opposite to angles A, B, and C respectively, then:

a² = b² + c² - 2bc cos(A).

By rearranging this formula based on the values we have, we can solve for the unknown side.

Imagine you're a surveyor trying to determine the distance between two locations that form a triangle with a known angle. By applying the Law of Cosines, you can calculate that unknown distance just like triangulating a position on a map using landmarks.

2. Using Law of Sines in SSA case

The Law of Sines is incredibly useful when solving for unknown sides or angles in cases where you know two sides and an angle that is not included between them (SSA). The Law of Sines formula is as follows:

a/sin(A) = b/sin(B) = c/sin(C).

This means that the ratio of the length of a side to the sine of its opposite angle is constant for all sides of the triangle, allowing you to find unknown angles or sides with some rearranging of the formula.

Think of it like using a pair of binoculars to spot a boat out on the water. If you know the angle to the boat and two distances (from yourself to two other known points), the Law of Sines helps you determine the distance to the boat, much like using triangulation to locate it on a map.

3. Constructing circum‑ and incircles with compass/straightedge

Circumcircles and incircles are important in triangle geometry. A circumcircle is the circle that passes through all the vertices of a triangle, while an incircle touches all the sides of the triangle. To construct these circles:

• For the circumcircle, find the perpendicular bisectors of at least two sides of the triangle. The intersection point of these bisectors gives you the center of the circumcircle. The radius can be measured from this center to any vertex.

• For the incircle, construct the angle bisectors of the triangle, where they intersect gives the incenter (the center of the incircle). The radius can then be determined by measuring the distance from the incenter to any side.

Imagine you are designing a pizza and want to put toppings evenly over it. The circumcircle is like the outer crust edge, ensuring your toppings cover the entire pizza. The incircle is like the area within the crust that you want to top deliciously, ensuring no toppings spill over the edges.

4. Proof: concurrency of medians at centroid (via coordinates or area method)

Medians of a triangle are segments that connect each vertex to the midpoint of the opposite side. The centroid is the point where all three medians intersect. To prove that these medians are concurrent (i.e., they meet at one point), you can either use a coordinate proof or an area argument. In a coordinate proof, you can assign coordinates to the triangle's vertices and demonstrate that the equations of the medians give the same intersection point. For area, you can show that the triangle is divided into smaller areas that balance at the centroid.

Picture a seesaw with three kids sitting at different points. The centroid acts like the balance point. Each median is a plank extending from the kids to the balancing point. Proving that all planks (medians) meet at the same spot explains why the seesaw balances. Each kid's weight creates a unique area contribution, ensuring equilibrium at the centroid.

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