Question

In arrangement e discuss How you can decompose the triangles so you can easily find the area of each triangle. Explain your method for finding the area of each triangle.

a) By using the Pythagorean theorem
b) Applying the law of cosines
c) Decomposing into known shapes and summing their areas
d) Using the formula for the area of a circle

Answer 1

To find the areas of triangles, one can use the Pythagorean theorem for right-angle triangles, the Law of Cosines for non-right-angled triangles, or decompose the triangle into known shapes. The formula for the area of a circle is not directly applicable to triangles.

Step-by-step explanation:

To decompose triangles and find their areas, one can use various methods depending on the information provided about the triangle. Here are a few strategies:

1. Pythagorean Theorem: In a right-angled triangle, we can use the Pythagorean theorem, a² + b² = c², to find the length of the hypotenuse (c) if we know the lengths of the other two sides (a and b). Once we have all side lengths, we can calculate the area using the formula ½ × base × height.
2. Law of Cosines: For non-right-angled triangles, we can use the Law of Cosines, c² = a² + b² - 2ab × cos(θ), to find the length of a side when we know the lengths of the other two sides and the included angle. With all three sides, we can use Heron's formula to find the area.
3. Decomposition: This involves breaking the triangle into known shapes such as rectangles, right triangles, or trapezoids, and then finding the area of each and summing them up to get the total area of the triangle.

Note that using the formula for the area of a circle is not typically applied to find the area of a triangle unless the triangle is inscribed within a circle, and even then, additional steps are required.