.jpg)
The formula for the area of triangles is very well known. Most high school students could recite that the area of a triangle is equal to one-half multiplied by the length of the base of the triangle multiplied by the length of the height of the triangle $A=\dfrac{bh}{2}$ One interesting aspect of triangles is that from their formula for area, we can derive the formula for the area of a square or rectangle. Every rectangle or square can be divided into two right triangles as shown in this example.
We know that $\triangle{ABC}\cong\triangle{CDB}$ by SSS congruence. Using Theorem 6 this means we know that the area of $\triangle{ABC}$ and the area of $\triangle{CDB}$ are equal. We also know the area of these two triangles summed together is equal to the triangle of our rectangle. Therefore, the area of the rectangle is $\dfrac{bh}{2}+\dfrac{bh}{2}=\dfrac{2bh}{2}=bh$. So the formula for the area of a rectangle is the length of the base times the length of the height.
Triangles can also be used to create the formula for the area of a circle. There are two different methods that triangles can be used in this process. Below are two apps from GeoGebra that will demonstrate how triangles help derive the formula for the area of a circle.
In the applet below you will see a circle divided into slices. We know that the area of the slices all summed together is equal to the area of the circle. If we reorganize the slices we see that the smaller the slices are made the more their reorganized form resembles a rectangle with a height of r and a length of $\pi$r. From the section above we know that we can find the area of a square by multiplying its height by its width, and we saw that this formula is derived using triangles. Therefore, as the slices are made infinitely small, their reorganized form will form a rectangle whose area is defined by $\pi{r^2}$.
Have you ever considered becoming a physicist, engineer, astronaut, cartographer, pilot, architect, surveyor, or even a crime scene investigator? If you enter into any of these professions, then you will use trigonometry extensively in your career. Trigonometry is based on right triangles and the relationships that right triangles have with their angles, hypotenuse, and legs. It is one of the most commonly used mathematical practices in the world. The GPS in your phone uses trigonometry to track distances and heights, astronomers frequently use trigonometry to map relationships between the stars and planets, and marine biologists use trigonometry to help track the movements and depths of pods of blue whales. Trigonometry's uses are limited only by the imagination of those using it, and without triangles, none of it would be possible. For more information about the uses of trigonometry, check out the links below.
It is no secret that triangles are one of the strongest shapes that exist. Due to their simplicity and strength, triangles are frequently used in the design and construction of homes, buildings, and bridges. The famous Golden Gate Bridge’s beams are built in the shape of a triangle to provide stability and strength, and most roofs are built in the shape of triangles to ensure a home is well protected from the elements. The congruence theorems discussed also play a part in understanding why triangles are so common in construction. Using these congruence theorems, construction companies only need to verify three elements of triangular beams are congruent to ensure the entire triangle will be congruent. This makes it easier to create level roofs, uniform bridges, and buildings that are not leaning like the Tower of Pisa, without forcing the workers to measure every aspect of each triangle to ensure its congruence.
10: Applications of Trigonometry. (2021, March 22). BYJUS. https://byjus.com/maths/applications-of-trigonometry/
11: Real life applications of trigonometry. (2017, October 16). Mathnasium. https://www.mathnasium.com/blog/real-life-applications-of-trigonometry
12: Or, A. (2018, October 2). Area of Cirles. GeoGebra. https://www.geogebra.org/m/dVXMR4U7
13: Brzezinski, T. (n.d.). Circle Area (By Peeling!). GeoGebra. Retrieved October 6, 2022, from https://www.geogebra.org/m/WFbyhq9d